JAN   S  191  i 
GIFT 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


GIFT    OF 


g   4  g 
•6 


A    TEXT-BOOK    OF 
PHYSICS  " 


BY 

S.    E.    COLEMAN,   S.B.,    A.M. 

HEAD  OF  THE  SCIENCE  DEPARTMENT  AND  TEACHER  OF  PHYSICS  IN  THE 
OAKLAND,  CALIFORNIA,  HIGH  SCHOOL;  AUTHOR  OF   "A  PHYSICAL 

LABORATORY  MANUAL,"  "NEW  LABORATORY  MANUAL  OF 
PHYSICS,"  AND  "THE  ELEMENTS  OF  PHYSICS" 


D.    C.   HEATH    &   COMPANY 

BOSTON  NEW    YORK  CHICAGO 


COPYRIGHT,  1911, 
BY  D.  C.   HEATH  &  Co. 


PREFACE 

THE  present  tendency  in  physics  teaching  is  to  attach 
less  importance  to  the  formal  and  academic  features  of  the 
subject,  and  to  lay  greater  stress  on  the  applications  of 
physics  in  daily  life.  This  change  of  front  is  in  accord  with 
the  general  movement  in  education  which  seeks  to  give 
the  subjects  of  instruction  a  more  useful  content,  drawn 
from  the  social  and  industrial  life  of  the  time.  Judged  by 
this  standard,  the  kind  of  physics  which  views  the  util- 
ities from  afar  or  ignores  them  altogether  is  discredited. 
But  a  word  of  caution  is  clearly  in  order.  A  new  truth 
is  never  the  whole  truth.  The  older  physics  had  much 
to  commend  it,  despite  the  caustic  criticisms  of  zealous 
reformers;  for  "the  most  practical  thing  in  the  world  is 
the  foundation  of  pure  science  upon  which  applied  sci- 
ence rests."  To  eliminate  or  minimize  the  fundamentals 
while  attempting  to  teach  their  applications  is  not  to  pro- 
vide a  "royal  road  to  learning,"  but  a  fool's  highway  to 
pretentious  ignorance.  Pure  and  applied  science  are 
equally  essential  to  a  well  rounded  course.  The  one  alone 
is  barren;  the  other,  when  not  well  founded  in  the  first,  is 
superficial,  disconnected,  and  trivial. 

A  first  course  in  physics  should  compass  results  which 
are  in  themselves  worth  while.  It  need  not  on  that  account 
be  any  the  less  valuable  as  a  preparation  for  college;  but 
a  course  which  recognizes  only  the  latter  goal  is  not  likely 
to  come  within  hailing  distance  of  any  other.  The  subject 
matter  should  be  drawn  largely  from  the  common  surround- 

iii 

227432 


iv  PREFACE 

ings  and  affairs  of  life,  about  which  the  pupil  already  knows 
something  and  about  which  it  concerns  and  interests  him 
to  know  more.  Scientific  education  should  begin  with  and 
develop  out; of  the  science  of  common  things. 

Genuine  knowledge  is  not  gained  by  the  contemplation 
of  laws  and  principles  in  the  abstract.  A  general  truth  can 
be  seen  only  through  the  medium  of  particular  instances. 
It  is  given  the  immature  student  to  see  the  great  generali- 
zations of  science  only  in  part,  at  the  best;  but  the  partial 
view  may  be  clear  and  vivid,  if  adequately  grounded  in 
experience.  The  premature  introduction  of  Newton's  laws 
of  motion,  presented  as  physical  axioms  (!)  and  buttressed 
mainly  by  formulas  and  problems  thrice  removed  from  the 
pupil's  experience,  has  only  served  to  bring  the  subject  of 
dynamics  into  disrepute.  The  principle  of  the  conserva- 
tion of  energy,  as  traditionally  presented,  is  a  further  ex- 
ample of  a  great  generalization  reduced  to  empty  verbiage 
—  a  sort  of  word  puzzle,  which  claims  the  attention  for  a 
moment  and  is  then  dismissed  for  good  and  all.  Hence  it 
is  that  elementary  physics  has  been  only  a  loose  aggrega- 
tion of  subjects,  having  little  apparent  relation  to  one 
another;  for  the  essential  unity  of  the  science  is  to  be  found 
only  in  the  idea  of  energy  and  its  conservation  in  all  physi- 
cal phenomena.  "The  doctrine  of  energy  plays  in  physical 
science  the  same  role  as  does  the  doctrine  of  evolution  in 
biological  science,  since  it  furnishes  concepts  and  a  termi- 
nology in  which  all  forms  of  physical  phenomena  may  be 
expressed.  This  terminology  and  these  concepts  are  partic- 
ularly useful,  because  they  are  derived  from  the  idea  of 
mechanical  work,  which  is  one  of  the  most  immediate  and 
familiar  of  the  concepts  drawn  from  daily  experience." 
Mechanical  principles  in  general  run  through  the  whole 
of  physics,  and  serve  as  the  necessary  basis  for  its  rational 
presentation.  While  it  does  not  follow  from  this  that  all 


PREFACE  V 

of  mechanics  must  come  first,  the  author  is  strongly  of  the 
opinion  that  this  is  the  best  plan.  It  is  true  that  many 
topics  in  mechanics  and  heat  are  so  closely  related  that  they 
may,  with  some  show  of  reason,  be  thrown  together.  But 
such  an  arrangement,  if  it  aims  at  correlation  (and  it  appar- 
ently has  no  other  warrant),  defeats  its  own  purpose;  for 
the  several  parts  of  mechanics  stand  in  a  more  intimate  and 
vital  relation  to  one  another  than  they  do  to  the  topics 
in  heat  or  any  other  branch  of  physics. 

Owing  to  the  diversity  which  exists  between  physics 
courses  planned  under  different  conditions,  the  scope  of  a 
text-book  written  for  general  use  is  necessarily  a  compro- 
mise. If  it  is  restricted  to  the  "essentials"  which  form  the 
common  ground  of  all  courses,  it  must  be  largely  supple- 
mented as  local  conditions  may  require.  If  its  scope  is 
broadened  to  include  a  more  complete  survey  of  phenomena 
and  principles  and  a  great  variety  of  industrial  applications, 
it  will  contain  more  than  should  be  attempted  with  most 
classes,  and  the  problem  of  the  teacher  will  be  to  select  that 
which  best  suits  his  purpose.  The  latter  type  of  book  is 
to  be  preferred  for  several  reasons.  In  the  first  place  it  is 
difficult  to  make  effective  use  of  reference  books  for  sup- 
plementary work,  especially  with  large  classes;  and,  at  best, 
this  is  a  time-consuming  expedient.  On  the  other  hand, 
supplementary  material  incorporated  in  the  text  itself  is 
ready  at  hand  when  wanted,  and  its  presence  invites  atten- 
tion and  stimulates  interest.  //  is  expected  that  the 
teacher,  in  using  this  book,  will  omit  considerable  portions 
at  his  discretion,  and  will  touch  lightly  on  other  portions 
which,  for  his  purpose,  are  to  be  regarded  as  of  minor 
importance.  To  cover  the  entire  course  in  detail  would 
require  a  year  and  a  half  with  most  classes. 

The  subject  matter  has  been  arranged  and  presented  with 
due  regard  to  its  correlation  with  the  laboratory  course. 


vi  PREFACE 

If  the  laboratory  experiments  are  to  have  any  particular 
value,  they  must  fit  into  the  general  scheme  of  the  text, 
just  as  the  class-room  experiments  do.  This  correlation 
has  been  worked  out  in  connection  with  the  author's  New 
Laboratory  Manual  of  Physics,  published  by  the  American 
Book  Company. 

This  preface  would  be  unduly  lengthened  were  any 
attempt  made  to  review  the  plan  of  the  book  in  detail  or 
the  treatment  of  the  different  subjects.  On  such  matters 
it  may  be  assumed  that  the  book  will  speak  intelligibly 
for  itself. 


CONTENTS 


PAGE 

CHAPTER    I.  INTRODUCTION .  I 

CHAPTER  II.  MATTER  AND  FORCE.     PHYSICAL  MEASUREMENTS 

I.   The  Three  States  of  Matter 7 

II.    Force  and  Inertia 9 

III.   Physical  Measurements 15 

CHAPTER  III.    STATICS  or  LIQUIDS 

I.    Introduction .      .      ...      .      .  22 

II.    Gravity  Pressure  in  Liquids 24 

III.  Transmission  of  Applied  Pressure  by  Liquids     ....  33 

IV.  Buoyancy  of  Liquids 36 

V.   Specific  Gravity 39 

CHAPTER  IV.     STATICS  OF  GASES 

I.   Atmospheric  Pressure 43 

II.   Laws  of  Gases 52 

III.   Applications  of  the  Mechanics  of  Fluids 61 

CHAPTER  V.    STATICS  or  SOLIDS 

I.    Concurrent  Forces 69 

II.   Parallel  Forces 78 

III.  Moments  of  Force 80 

IV.  Effect  of  Weight  on  the  Equilibrium  of  Bodies  ....  84 
V.   Elasticity.     Stresses  and  Strains 92 

CHAPTER  VI.    DYNAMICS 

I.   Motion 104 

II.   Newton's  Laws  of  Motion 120 

III.  The  Laws  of  Motion  in  Special  Cases 135 

IV.  Work  and  Kinetic  Energy 150 

V.   Machines 159 

VI.   Energy 176 

VII.    Dynamics  of  Fluids 183 

vii 


viii  PHYSICS 

PAGE 

CHAPTER  VII.    THE  MOLECULAR  THEORY  OF  MATTER 

I.   The  Structure  of  Matter 196 

II.   Molecular  Properties  of  Gases 202 

III.  Molecular  Properties  of  Liquids 207 

IV.  Molecular  Properties  of  Solids .      .  214 

CHAPTER  VIII.    HEAT 

I.   Nature  of  Heat 218 

II.   Temperature. 220 

III.  Conduction  and  Convection 225 

IV.  Radiation 230 

V.    Changes  in  Volume  and  Pressure 240 

VI.   Measurement  of  Heat.     Specific  Heat 248 

VII.   Fusion  and  Solidification 252 

VIII.   Vaporization  and  Condensation 260 

IX.   Heating  and  Ventilation  of  Buildings 283 

X.   Heat  and  Other  Forms  of  Energy 287 

XI.   Heat  Engines 292 

CHAPTER  IX.     SOUND 

I.   Origin  and  Transmission  of  Sound      .            306 

II.   Properties  of  Musical  Sounds 322 

III.   Sympathetic  and  Forced  Vibrations.     Resonance    .      .      .  342 

CHAPTER  X.    LIGHT 

I.   Nature  and  Transmission  of  Light 359 

II.   Intensity  of  Illumination.     Candle  Power 368 

III.  Reflection  of  Light        .      .     .     .     .     .     .     ...     .  372 

IV.  Refraction  of  Light       .     .     ...     .     .     .     ..  *.     .  389 

V.  Lenses 405 

VI.   The  Eye ;........  416 

VII.   Optical  Instruments 425 

VIII.   Dispersion  of  Light.     Color 437 

CHAPTER  XL    MAGNETISM 

I.   Properties  of  Magnets 457 

II.   The  Magnetic  Field       .     .     .     .     .     .     ...     .     .  466 

III.  The  Earth's  Magnetic  Field 472 


PHYSICS  ix 

PAGE 

CHAPTER  XII.     ELECTROSTATICS 477 

CHAPTER  XIII.     ELECTRODYNAMICS 

I.   Introduction 501 

II.   Primary  Cells :  503 

III.  The  Magnetic  Action  of  a  Current 517 

IV.  Measurement  of  Electric  Currents 531 

V.   Ohm's  Law 537 

VI.   Laws  of  Resistence 541 

VII.   Measurement  of  Resistance  and  Electromotive  Force  .      .  545 

VIII.   Electrical  Energy.    Heating  Effects  of  Electric  Currents    .  553 

IX.   Elctromagnetic  Induction 564 

X.   Chemical  Effects  of  the  Electric  Current 600 

CHAPTER  XIV.    RADIATIONS.  THE  ELECTRICAL  NATURE  OF  MATTER 

I.    Spectra  and  Spectrum  Analysis 610 

II.   Electric  Oscillations  and  Waves.     Electromagnetic  Theory 

of-Light 617 

III.  Electric  Conduction  through  Gases.    Cathode  and  Rontgen 

Rays 622 

IV.  Radioactivity.     Electrical  Theory  of  Matter     ....  630 

APPENDIX 638 

INDEX 641 


A  TEXT-BOOK 
OF    PHYSICS 


A  TEXT-BOOK  OF  PHYSICS 


CHAPTER  I 
INTRODUCTION 

1.  Scientific  Education. — From  infancy  a  child  is  busy 
acquiring  facts  concerning  the  strange  world  in  which  he 
finds  himself.  It  matters  little  what  may  chance  to  turn 
up,  he  wants  to  scrape  an  acquaintance  with  it,  whether 
it  is  a  bug  or  a  circus  elephant,  a  jumping-jack  or  a  steam- 
engine.  So  earnest  is  he  about  this  business  that  he  be- 
sieges his  elders  with  questions,  and  fairly  earns  the  title 
of  the  human  interrogation-point.  He  absorbs  miscel- 
laneous information  as  naturally  as  a  sponge  absorbs  water. 
In  his  earlier  years  the  boy's  interest  is  awakened  only  by 
the  things  that  appeal  directly  to  his  senses.  He  is  intent 
upon  seeing,  feeling,  hearing,  and  tasting.  He  wants  to 
know  what  this  or  that  thing  is  called,  what  it  does,  and 
what  it  is  for. 

In  all  this  the  child  is  taking  the  first  step  toward  scien- 
tific knowledge.  The  first  step  does  not  carry  him  far, 
to  be  sure,  but  it  is  a  necessary  step.  The  boy  is  gathering 
the  raw  materials  of  science,  not  with  any  thought  of  their 
possible  future  value,  but  just  because  he  wants  to  know. 
During  these  early  years  the  collection  of 'facts  stored  up 
in  his  small  brain  resembles  a  scrap-heap  of  odds  and  ends 
rather  than  a  well  kept  museum.  By  and  by  he  begins 


2  "...•          v    •  A'.  £HYSICS 

•  »  •     •  *» 

to  fat ; his'  >p±ellecjual -house -in  order.     This  is  the  second 

step  in  his  scientific  education,  and,  like  the  first,  it  is  taken 
without  conscious  purpose.  Some  day,  perhaps,  it  occurs 
to  him  that  the  flight  of  his  kite  is  not  altogether  unlike  the 
flight  of  a  bird,  and  he  begins  to  compare  their  behavior. 
Both  can  remain  in  the  air  indefinitely,  although  objects 
in  general  fall  to  the  ground  unless  they  have  some  visible 
support.  Light  objects  are  borne  about  by  the  wind  and 
remain  in  the  air  for  some  time,  and  the  kite  also  requires 
a  wind;  but  a  bird  can  fly  in  still  air,  although  its  body 
may  be  large  and  heavy.  A  bird  evidently  keeps  from  fall- 
ing and  maintains  its  flight  by  flapping  its  wings;  but  how 
does  the  motion  of  the  wings  bring  about  this  result?  Here 
is  the  puzzle,  and  the  boy  pokes  about  meditatively  among 
his  scrap-heap  of  facts  for  something  to  match  it.  Ah, 
here  it  is!  Flying  must  be  something  like  swimming. 
Swimming  is  a  part  of  the  boy's  personal  experience,  and 
he  knows  that  to  drive  his  body  forward  he  must  push 
back  on  the  water  with  his  hands  and  feet.  Then  it  must 
be  that  the  swift  downward  stroke  of  the  bird's  wings 
against  the  air  gives  its  body  an  upward  impulse  which 
keeps  it  from  falling.  In  some  way,  probably  due  to  the 
shape  of  the  wings,  the  impulse  is  partly  forward  as  well 
as  upward;  otherwise  there  would  be  no  forward  motion. 
Speculating  thus  about  flying,  the  boy  recalls  that  he  has 
seen  a  balloon  rise  in  the  air  without  the  aid  of  a  wind  and 
without  wings  to  flap.  Evidently  it  is  neither  like  a  kite 
nor  a  bird.  Here,  then,  is  a  new  problem:  To  find  out 
what  makes  a  balloon  rise. 

By  the  time  a  boy  reaches  the  age  of  twelve  or  fourteen 
years  his  head  is  full  of  such  problems.  No  sooner  is  one 
solved  to  his  satisfaction  than  others  come  to  take  its 
place.  He  has  outgrown  the  chance  world  of  his  earlier 


INTRODUCTION  3 

years,  in  which  the  things  that  happen  just  happen,  and  is 
beginning  to  realize  that  there  is  a  connected  scheme  of 
things,  or  plan,  in  which  everything  that  happens  has  its 
proper  place.  His  interest  is  now  centered  in  the  how  and 
the  why  of  things.  How  does  the  engine  work?  How 
does  the  phonograph  talk,  or  the  telephone?  Why  do 
the  hills  mock  him  with  their  echo?  How  does  the  lens  of 
his  camera  make  the  picture  on  the  film?  Why  does  his 
gun  "kick"  when  he  fires  a  shot?  What  keeps  the  people 
on  the  under  side  of  the  earth  from  falling  off?  What  makes 
the  moon  change  from  crescent  to  quarter  and  from  quar- 
ter to  full  moon  every  month?  What  makes  the  winds? 
And  so  on  without  end. 

By  dint  of  thinking  and  asking  questions  the  boy  arrives 
at  some  sort  of  answer  to  his  problems.  Thus  in  a  rather 
aimless  fashion  he  is  engaged  in  sorting  over  the  facts  of 
his  experience,  comparing,  classifying,  and  drawing  gen- 
eral conclusions  which  may  be  of  use  to  him.  As  a  rule, 
however,  his  conclusions  are  vague  and  inaccurate,  for  his 
acquaintance  with  facts  is  limited  and  he  is  not  a  trained 
thinker.  He  has  made  a  good  start  in  his  scientific  educa- 
tion, and  has  made  it  in  the  natural  way;  but  this  is  about 
as  far  as  the  haphazard  experiences  and  interests  of  daily 
life  will  carry  him.  The  next  step  in  advance  demands 
systematic,  purposeful  effort,  directed  toward  a  definite 
end.  The  opportunity  for  such  effort  is  afforded  by  the 
science  courses  of  school  and  college.  In  the  pursuit  of 
any  branch  of  science  the  student's  incomplete  and  frag- 
mentary acquaintance  with  the  main  facts  of  the  science 
is  pieced  out  by  observation  and  experiment  in  the  labora- 
tory and  the  class-room,  under  the  guidance  of  the  teacher. 
Having  this  substantial  acquaintance  with  a  wide  range 
of  facts  at  first  hand  the  student  will  be  able  to  make  effect- 


4  PHYSICS 

ive  use  of  a  text-book  in  gaining  a  more  comprehensive 
knowledge  of  the  subject  than  would  be  possible  through 
his  personal  observation  and  experience  alone.  The 
teacher,  the  laboratory,  and  the  text-book  are  all  essential 
to  the  best  results. 

2.   Physics  and  its  Place  among  the  Sciences.  —  The 

knowledge  of  the  material  universe  is  subdivided,  for  con- 
venience, into  several  branches,  called  the  natural  sciences. 
The  biological  sciences  treat  of  living  things;  the  physical 
sciences  deal  with  inanimate  matter  in  all  its  forms,  and 
with  the  changes  and  processes  which  it  undergoes.  The 
physical  sciences  are  physics,  chemistry,  astronomy, 
geology,  meteorology,  and  mineralogy. 

Physics  is  the  broadest  of  the  natural  sciences  and  shares 
with  chemistry  the  honor  of  being  the  foundation  of  all 
the  others.  All  changes,  occurrences,  or  processes  which 
take  place  in  the  material  world  are  of  either  a  physical  or 
a  chemical  nature.  Any  action  or  process  in  which  matter 
changes  from  one  kind  to  another  is  a  chemical  process. 
Combustion  or  burning  is  the  most  familiar  example. 
The  substance  burned  unites  with  oxygen  from  the  air, 
forming  certain  gases  which  pass  off  into  the  air.  The 
rusting  of  iron  and  the  decay  of  animal  and  vegetable 
matter  are  further  examples.  A  physical  process  is  one 
that  does  not  involve  a  change  of  matter  from  one  kind 
to  another;  e.g.  the  melting  of  ice  and  the  evaporation  of. 
water.  Water  is  the  same  substance  whether  it  exists  as 
ice,  liquid  water,  or  water  vapor.  A  change  from  one  to 
the  other  is  a  change  of  state,  or  of  physical  condition. 
It  is  this  distinction  between  chemical  and  physical  change 
which  serves  as  the  dividing  line  between  chemistry  and 
physics. 


INTRODUCTION  5 

The  five  great  departments  of  physics  are  Mechanics, 
Heat,  Sound,  Light,  and  Electricity,  the  last  including  the 
closely  related  subject  of  Magnetism.  Mechanics  treats 
of  the  action  of  forces  in  determining  the  state  of  rest  or 
motion  of  bodies.  It  presents  the  fundamental  principles 
which  are  applied  in  the  construction  and  use  of  machines. 
A  boy  in  learning  the  control  of  his  own  body  in  walking, 
running,  jumping,  balancing,  alighting  from  a  moving  car, 
riding  a  bicycle,  etc.,  is  gaining  experience  in  mechanical 
matters.  Mechanics  is  the  fundamental  branch  of  physics. 
Its  general  principles  run  as  a  network  through  all  depart- 
ments of  the  science.  The  phenomena*  of  heat,  sound, 
light,  and  electricity  with  which  every  one  becomes 
acquainted  through  the  experiences  of  daily  life  will  serve, 
in  a  general  way,  to  indicate  the  field  covered  by  these 
branches. 

3.  Physics  as  a  Study.  —  Physics,  rightly  studied,  is 
not  a  burden  to  the  memory  but  an  aid  to  the  understand- 
ing. The  facts  of  physics  are  easily  remembered  when  they 
are  understood;  if  they  are  not  understood,  it  matters  little 
whether  they  are  remembered  or  not. 

Information  is  valuable  as  an  intellectual  possession, 
but  the  ability  to  think  accurately  is  of  much  greater  value. 
The  student  of  physics  has  at  his  disposal  one  of  the  best 
means  of  developing  this  ability. 

A  fact  clothed  in  slovenly  or  ambiguous  phrase  cuts  a 
sorry  figure.  Physics  is  an  exact  science  and  finds  suitable 
expression  only  in  exact  speech.  Training  in  the  use  of 

*  A  phenomenon,  as  the  term  is  used  in  science,  is  any  action  or  occur- 
rence perceived  by  the  senses,  however  familiar  and  commonplace  it  may 
be.  The  visible  happenings  in  nature  are  collectively  termed  natural  phe- 
nomena; and  if  they  involve  only  physical  processes,  they  are  further 
classified  as  physical  phenomena. 


6  PHYSICS 

the  mother  tongue  is  one  of  the  lasting  benefits  to  be  derived 
from  the  study  of  the  subject. 

The  material  results  of  science  and  invention  are  about 
us  on  every  hand.  To  be  able  to  use  them  is  practically 
a  necessity;  to  understand  them,  in  some  measure  at  least, 
is  a  necessary  part  of  a  liberal  education;  to  add  to  them 
is  to  contribute  something  toward  human  progress.  A 
knowledge  of  physics  is  a  valuable  aid  toward  all  these 
ends. 


CHAPTER  II 

MATTER  AND  FORCE.    PHYSICAL  MEASUREMENTS 
I.  THE  THREE  STATES  OF  MATTER 

4.  Matter  exists  in  three  physical  states  or  conditions, 
called  the  solid,  the  liquid,  and.  the  gaseous  states.  Much 
of  physics  depends  upon  the  characteristic  properties  which 
distinguish  the  states  of  matter  from  one  another.  Thus 
we  have  the  mechanics  of  solids,  the  mechanics  of  liquids, 
and  the  mechanics  of  gases.  These  properties,  therefore, 
require  some  attention  at  the  outset. 

Liquids  are  distinguished  from  solids  by  the  fact  that 
they  tend  to  flow,  and  hence  must  be  contained  in  vessels. 
Every  solid,  on  the  contrary,  has  a  shape  of  its  own,  which 
it  tends  to  preserve.  Some  solids,  e.g.  stone  and  iron, 
offer  great  resistance  to  a  change  of  shape;  others,  such  as 
wet  clay  and  putty,  can  readily  be  molded  into  any  form. 
But  even  the  small  amount  of  resistance  offered  by  soft 
solids  distinguishes  them  from  liquids. 

Many  of  the  physical  properties  of  gases  may  be  learned 
from  a  study  of  the  air,  which  is  a  mixture  of  several  gases, 
principally  nitrogen  and  oxygen.  Although  the  air  is 
everywhere  about  us,  we  are  ordinarily  unconscious  of  its 
existence  unless  it  is  in  motion.  When  it  is  in  motion,  we 
recognize  it  as  a  current  of  air,  a  breeze,  or  a  wind.  We 
commonly  call  a  vessel  "empty"  when  it  is  full  of  air;  and 
seldom  stop  to  think  that  when  the  so-called  empty  ves- 
sel is  being  filled  with  a  liquid  or  a  solid,  the  air  in  it  is  being 
pushed  out. 

7 


8  MATTER    AND    FORCE 

It  will  help  toward  clear  thinking  on  this  point  to  push  an  inverted 
tumbler  into  a  vessel  of  water.  The  water  does  not  rise  to  fill  the 
tumbler,  being  prevented  from  doing  so  by  the  confined  air;  but, 
when  the  tumbler  is  slowly  inclined,  the  air  escapes  in  a  succession 
of  bubbles,  and  the  water  enters  at  the  same  time  to  take  its  place. 

This  simple  experiment  shows  that  a  body  of  air  confined  in  any 
space  tends  to  keep  other  bodies  out  of  that  space;  and  the  same  is 
true  of  all  gases.  But  we  know  that,  after  a  bicycle  or  an  automobile 
tire  is  fully  inflated,  much  air  must  still  be  pumped  in  to  make  it 
hard.  Now  air  can  be  forced  into  the  fully  inflated  tire  only  by  com- 
pressing the  air  already  in  it  into  a  smaller  space;  and  experience 
teaches  that  the  compression  of  the  confined  air  can  be  carried  as 
far  as  the  strength  of  the  tire  or  of  the  operator  will  permit.  The 
great  compressibility  of  air  can  be  shown  simply  by  pushing  in  the 
piston  of  a  bicycle  pump  or  other  compression  pump,  while  the  outlet 
is  closed  with  the  finger.  A  vigorous  push  will  compress  the  air  per- 
haps to  one  half  or  even  to  one  third  of  its  original  volume.  When 
the  piston  is  released,  the  air  expands  and  drives  it  back. 

All  gases  are  highly  compressible  and  expansible,  like 
air.  When  any  quantity  of  gas,  however  small,  is  ad- 
mitted into  an  otherwise  empty  space  it  instantly  expands 
so  as  to  fill  the  space  completely. 

If  the  above  experiment  is  repeated  with  the  com- 
pression pump  filled  with  water,  it  will  be  found  that 
the  water  is  as  unyielding  as  a  board,  for  the  piston 
cannot  be  pushed  in  at  all. 

All  liquids  and  most  solids  are  only  very  slightly  com- 
pressible. Even  under  very  great  pressure  their  change 
of  volume  is  commonly  so  slight  as  to  escape  notice;  and, 
for  all  practical  purposes,  they  are  regarded  as  incompres- 
sible. Hence  great  compressibility  and  expansibility  are 
distinguishing  properties  of  the  gaseous  state. 

5.  Summary.  —  A  solid  tends  to  preserve  a  definite 
shape  and  volume.  A  liquid  tends  to  preserve  a  definite 
volume,  but  has  no  shape  of  its  own,  since  its  parts  move 


FORCE  AND  INERTIA  9 

readily  over  one  another.  A  gas  has  neither  a  self-deter- 
mining shape  nor  volume;  it  is  highly  compressible,  and 
tends  to  expand  indefinitely. 

Since  both  liquids  and  gases  flow,  they  are  classed  together 
as  fluids.  The  fluidity  of  gases  can  be  shown  in  an  inter- 
esting way  by  pouring  carbonic-acid  gas  upon  a  lighted  can- 
dle, from  a  jar  filled  with  the  gas,  just  as  water  is  poured 
from  a  vessel.  The  gas,  being  considerably  heavier  than  air, 
falls  in  a  stream  upon  the  candle  and  extinguishes  it. 

6.  Changes  of  State.    Vapors.  —  The  gaseous  form  of  a 
substance  which  exists  as  a  liquid  at  ordinary  tempera- 
tures is  called  a  vapor.     Many  substances  exist  in  two  or  all 
of  the  physical  states.     Water  is  a  familiar  example.     The 
metals  and  some  other  solids  can  be  liquefied  and  vaporized 
by  the  application  of  heat.     Some  solids,  such  as  wood, 
undergo  a  chemical  change  with  the  application  of  heat, 
instead  of  a  change  of  state. 

II.    FORCE  AND  INERTIA 

7.  Force.  —  The  word  force,  as  used  in  physics,  is  a  gen- 
eral term  for  any  push  or  pull.     The  following  are  familiar 
examples  of  forces:  the  pull  exerted  by  a  horse  upon  a 
wagon;  the  push  or  pull  by  which  a  door  is  opened;  the 
strong  push  or  pressure  exerted  by  a  hammer  at  the  in- 
stant it  strikes  a  nail;  the  downward  pressure  of  a  book 
against  a  table  upon  which  it  is  lying  and  the  upward  sus- 
taining pressure  of  the  table  against  the  under  side  of  the 
book;  the  pressure  of  a  liquid  against  the  bottom  and  sides 
of  the  containing  vessel. 

8.  Inertia.  —  We  learn  from  daily  experience  that  a 
body  acquires  motion  only  as  the  result  of  an  applied  force, 
exerted  upon  it  by  some  other  body.     For  example,  a  ball 
is  sent  flying  through  the  air  by  a  vigorous  push  of  the  hand 


10 


MATTER    AND    FORCE 


in  the  act  of  throwing  it,  or  by  a  blow  with  a  bat;  a  high 
velocity  is  imparted  to  a  rifle-ball  by  the  pressure  of  the 
gases  from  the  powder  exploded  behind  it;  and  a  wagon  is 
started  by  the  pull  exerted  by  the  horses  upon  it  through 
the  traces. 

It  is  also  a  matter  of  common  observation  that  moving 
bodies  come  to  rest  more  or  less  slowly  after  the  forces 
that  start  them  cease  to  act.  A  book  slides  over  a  table 
when  started  with  a  sudden  push,  but  quickly  stops;  a  ball 
can  be  made  to  roll  a  long  distance  over  a  smooth,  level 
surface,  as  a  sidewalk,  but  gradually  loses  speed  till  it  comes 
to  rest;  and  a  wagon  goes  only  a  short  distance  after  the 
horses  cease  to  pull  This  behavior  of  moving  bodies  is 
not  due  to  any  tendency  of  the  bodies  themselves  to  come 
to  rest,  but  is  the  effect  of  opposing  forces  which  are  devel- 
oped by  the  rubbing  of  one  surface  over  another.  Such  a 
force  is  called  friction.  Friction  acts  as  a  resistance  to 
motion,  and  tends  to  bring  moving  bodies  to  rest.  The 
smoother  the  surfaces  are,  the  less  friction  becomes;  hence 

a  body  slides  farther  on 
a  smooth  surface  than 
on  a  rough  one.  A 
skater,  for  example,  can 
go  a  long  distance  with- 
out effort  after  getting 
up  speed,  the  friction 
between  skates  and  smooth  ice  being  very  light.  Rolling 
friction  is,  in  general,  much  less  than  sliding  friction;  hence 
the  use  of  wheels  on  vehicles  of  all  sorts.  Ball  bearings 
(Fig.  i)  reduce  friction  still  further  by  substituting  rolling 
friction  for  sliding  friction  at  the  axle. 

Another  hindrance  to  motion  is  the  resistance  of  the 
air.  This  resistance  is  small  upon  a  body  moving  slowly, 


FIG.  i.— Ball  Bearings. 


FORCE  AND  INERTIA  n 

but  rapidly  increases  with  the  velocity.  For  high  veloci- 
ties, such  as  those  of  an  express-train  or  a  rifle-ball,  it  is 
very  great.  Bodies  can,  of  course,  be  stopped  by  other 
forces  than  friction.  A  ball  is  stopped,  when  caught,  by 
the  pressure  of  the  hands  against  it. 

The  general  truth  to  be  gathered  from  such  facts  as 
the  above  is  that  the  existing  state  of  rest  or  motion  of 
a  body  can  be  changed  only  by  means  of  a  force  of  some 
sort  acting  upon  the  body  from  without;  in  other  words, 
a  body  can  not  of  itself  alone  move,  if  at  rest,  or  change 
its  motion,  if  moving.  This  is  true  of  all  matter,  solid, 
liquid,  and  gaseous,  animate  and  inanimate.  Matter  of 
itself  tends  to  continue  in  whatever  state  of  rest  or 
motion  it  may  chance  at  that  instant  to  be.  This  prop- 
erty of  passiveness  is  called  the  inertia  of  matter,  or 
simply  inertia. 

9.  Action  of  Forces  With  and  Without  Contact.  Weight. 
—  All  of  the  forces  previously  mentioned  are  exerted  by 
direct  or  indirect  contact  of  the  body  exerting  the  force  and 
the  body  upon  which  the  force  is  exerted.  Thus  a  horse  in 
drawing  a  wagon  pushes  on  the  collar  with  his  shoulders, 
the  collar  pulls  on  the  traces,  and  the  traces  pull  on  the 
wagon.  Certain  forces,  however,  act  without  any  material 
connection  between  the  bodies  concerned.  The  forces 
exerted  by  a  magnet  are  of  this  sort.  Pieces  of  iron  move 
toward  a  magnet  and  cling  to  it.  We  know  from  this 
behavior  of  the  iron  that  it  is  acted  upon  by  a  force  whose 
direction  is  toward  the  magnet,  although  there  is  nothing 
whatever  to  show  how  this  force  is  exerted. 

Similarly,  the  fact  that  an  unsupported  body  falls  indi- 
cates that  a  downward  force  is  acting  on  it.  This  force 
is  in  some  unknown  way  due  to  the  earth,  and  we  think  of 
the  earth  as  exerting  a  pull  or  attraction,  by  which  it  tends 


12  MATTER    AND    FORCE 

to  draw  all  bodies  toward  its  center.  The  attraction  ex- 
erted by  the  earth  upon  any  body  is  called  the  weight  of 
the  body.  (As  a  result  of  the  earth's  rotation  on  its  axis, 
the  weight  of  a  body  is  very  slightly  less  than  the  earth's 
attraction  for  it,  except  at  the  poles.) 

Forces  such  as  those  exerted  by  a  magnet  or  by  the  earth 
are  none  the  less  real  because  we  do  not  know  how  they  act. 
The  muscular  sensation  of  effort  which  we  experience  in 
resisting  these  forces  is  convincing  evidence  that  they  are 
real.  A  piece  of  iron,  when  held  near  a  strong  magnet, 
pulls  the  hand  toward  the  magnet  with  a  force  that  is  eas- 
ily felt.  The  iron  pulls  the  hand  because  it  is  itself  pulled. 
Similarly,  when  we  lift  any  object  it  exerts  a  downward 
pull  upon  the  hand  and  arm,  which  it  does  only  because 
there  is  an  equal  downward  force  (its  weight)  acting  on  it. 

10.  Balanced  and  Unbalanced  Forces.  —  A  force  acting 
alone  on  a  body  always  sets  it  in  motion  or  changes  its 
existing  motion.  A  stone  flying  through  the  air  affords  a 
good  example;  for  its  weight  is  practically  the  only  force 
acting  on  it  during  its  flight,  the  resistance  of  the  air  being 
very  small.  The  weight  of  the  stone  causes  a  continuous 
decrease  of  speed  if  the  stone  is  rising  vertically,  a  contin- 
uous increase  of  speed  if  it  is  falling  vertically,  and  a  con- 
tinuous change  of  both  speed  and  direction  if  it  is  moving 
obliquely. 

Two  or  more  forces  acting  on  a  body  at  the  same  time 
may  be  so  opposed  to  each  other  that  the  body  behaves 
exactly  as  if  none  of  the  forces  were  acting.  Such  forces 
are  said  to  balance  each  other,  or  to  be  in  equilibrium,  and 
are  called  balanced  forces.  For  example,  when  two  boys 
pull  equally  and  in  opposite  directions  on  a  cart,  the  two 
pulls  are  in  equilibrium,  and  the  cart  remains  at  rest. 
Similarly,  the  weight  of  a  body,  at  rest  or  in  motion  on  a 


FORCE  AND  INERTIA  13 

level  surface,  is  sustained  or  balanced  by  the  upward  pres- 
sure of  the  surface;  and  the  weight  of  a  body  suspended 
by  a  cord  is  balanced  by  the  upward  pull  of  the  cord.  In 
both  cases  the  sustaining  force  is  equal  and  opposite  to 
the  weight  of  the  body.  These  examples  illustrate  the 
simplest  case  of  balanced  forces,  namely,  that  of  two  equal 
forces  acting  in  opposite  directions  along 
the  same  line.  An  example  of  three 
forces  in  equilibrium  is  shown  in  Fig.  2. 

The  body  is  supported  by  two  cords. 
Each  cord  pulls  obliquely  upward  on  the 
body,  and  these  two  pulls  together  bal- 
ance the  weight  of  the  body. 

A  single  force  acting  on  a  body  is  necessarily  an  unbal- 
anced force;  two  or  more  forces  acting  together  may  be 
either  balanced  or  unbalanced,  depending,  in  part,  upon 
their  relative  directions.  These  matters  are  to  be  studied 
further  in  the  following  chapters. 

11.  Resultant  Force.  —  In  many  cases  where  two  or 
more  forces  act  at  the  same  time  upon  a  body  they  affect 
its  behavior  (as  regards  rest  or  motion)  exactly  as  some  one 
force  would .  This  sm^l(e_^qui valeri t Jorce.  is  .called  the  result- 
ant  of  the  given  forces.  For  example,  if  a  boy  pulls  on  a 
cart  with  a  force  of  1 5  pounds,  and  another  boy  pulls  with 
him,  exerting  a  force  of  25  pounds,  the  effect  on  the  cart  will 
be  the  same  as  if  one  boy  alone  pulled  with  a  force  of  40 
pounds  in  the  same  direction.  In  general,  the  resultant  of 
any  number  of  forces  which  act  along  the  same  line  and  in 
the  same  direction  is  a  force  equal  to  their  sum,  acting  along 
the  same  line  and  in  the  same  direction  as  the  given  forces. 

The  resultant  of  two  forces  which  act  in  opposite  direc- 
tions along  the  same  line  is  a  force  equal  to  their  difference, 
acting  along  the  same  line  and  in  the  direction  of  the 


14  MATTER    AND    FORCE 

greater.  Thus  if  the  forces  exerted  on  the  cart  in  the  above 
example  are  in  opposite  directions,  they  will  together  be 
equivalent  to  a  single  force  of  10  pounds  acting  in  the 
direction  of  the  25-pound  pull., 

The  resultant  of  two  equal  forces  acting  in  opposite  direc- 
tions along  the  same  line  is  zero,,  since  the  two  forces  ex- 
actly neutralize  each  other.  The  resultant  of  any  set  of 
balanced  forces  is  zero,  for  the  same  reason. 

12.  The  Mutual  Action  of  Two  Bodies. — Force  is  always 
a  two-sided  action.  Whenever  one  body  exerts  a  force 
on  another,  the  second  body  exerts  at  the  same  time  an 
equal  and  opposite  force  on  the  first.  This  is  often  evident 
from  the  effects  produced.  For  example,  when  one  marble 
strikes  another,  it  sets  that  one  in  motion,  and  is  itself 
stopped  or  retarded  by  the  opposite 
force  which  the  second  marble  ex- 
erts on  it.  The  mutual  action  of  a 
ball  and  a  bat  is  a  similar  case. 
When  a  bullet  strikes  a  board  the 
force  it  exerts  makes  a  hole  in  the 
board;  the  equal  and  opposite  force 
exerted  on  trie  bullet  by  the  board 
stops  the  bullet. 
FIG.  3.  Since  these  familiar  examples  fur- 

nish no  direct  evidence  that  the  forces  exerted  by  two 
bodies  on  each  other  are  equal,  it  will  be  instructive  to 
try  an  experiment  especially  contrived  to  show  this  fact. 

The  apparatus  consists  of  two  hardwood  or  ivory  balls,  sus- 
pended as  shown  in  Fig.  3.  One  of  the  balls  is  drawn  aside  and 
released.  It  falls,  strikes  the  other  ball,  and  instantly  stops;  while 
the  other  swings  out  as  far  (very  nearly)  as  the  first  ball  would  have 
gone  if  its  motion  had  not  been  hindered.  Since  the  balls  are  exactly 
alike  and  the  one  loses  as  much  motion  as  the  other  gains,  it  follows 


PHYSICAL  MEASUREMENTS  15 

that  the  forces  which  they  exert  on  each  other  are  equal.  (If  the  balls 
were  of  unequal  size  they  would  still  exert  equal  forces  on  each  other; 
but  the  proof  of  this  involves  matters  not  yet  considered.) 

The  forces  exerted  between  two  bodies  at  rest  are  also 
equal  and  opposite.  Thus,  when  the  hand  is  pressed  against 
a  wall,  the  wall  presses  back  on  the  hand  with  equal  force. 
A  book  lying  on  a  table  exerts  a  downward  pressure  equal 
to  its  weight;  the  resistance  offered  by  the  table  acts  as  an 
equal  upward  pressure  on  the  book. 

PROBLEMS 

1.  Discuss  any  phenomena  with  which  you  are  familiar  that  show  the 
inertia  of  water;   the  inertia  of  air;   the  inertia  of  your  own  body* 

2.  In  what  direction  is  an  inexperienced  person  likely  to  fall  on  alighting 
from  a  rapidly  moving  car?     Why? 

3.  Discuss  some  good  example  of  balanced  forces.  Of  unbalanced  forces. 

4.  What  forces  are  acting  on  a  wagon  when  drawn  at  a  uniform  rate  on 
a  level  road?     Are  they  balanced  or  unbalanced? 

5.  A  boy  exerts  a  lifting  force  of  75  Ib.  on  a  stone  weighing  200  Ib.    (a)  Is 
this  a  balanced  or  an  unbalanced  force?     (6)  What  balanced  forces  are  act- 
ing on  the  stone? 

6.  Is  it  the  forces  exerted  by  or  upon  a  body  that  affect  its  state  of  rest 
or  motion? 

7.  Make  a  list  of  any  phenomena  which  seem  to  you  to  indicate  (a)  that 
some  bodies  are  without  inertia;  (&)  that  there  is  matter  which  has  no  weight; 
(c)  that  any  body  can  exert  a  force  on  another  without  the  other  exerting  at 
the  same  time  an  equal  and  opposite  force  on  it.    If  you  find  any  such  seeming 
exceptions  to  the  statements  made  in  the  text,  save  the  list  for  future  study. 

III.     PHYSICAL  MEASUREMENTS 

13.  Measurement  and  Units  of  Measurement.  —  Experi- 
mental work  in  physics  consists  largely  in  measuring  the 
different  kinds  of  physical  quantities,  such  as  length,  sur- 
face, volume,  force,  velocity,  time,  mass,  etc.  Any  kind  of 
quantity  is  measured  by  finding^  how  many  times  it  con- 
tains a  certain  fixed  or  standard  amount  of  that  kind  of 
quantity.  This  standard  amount  is  called  a  unit;  and  there 


16  MATTER    AND    FORCE 

are  various  units  in  common  use  for  measuring  each  kind 
of  quantity.  Thus  for  measuring  length  we  have  the  inch, 
foot,  meter,  centimeter,  etc. 

On  account  of  the  great  simplicity  of  the  metric  system 
of  measures,  it  is  almost  exclusively  used  in  scientific  work. 
It  is  the  only  system  that  we  need  consider  here. 

14.  Units  of  Extension.  —  The  primary  unit  of  length 
in  the  metric  system  is  the  meter.  It  is  defined  as  the  dis- 
tance between  two  lines  on  a  certain  metallic  rod  preserved 
in  the  archives  of  the  International  Metric  Commission, 
in  Paris,  the  rod  being  at  the  temperature  of  melting  ice. 
This  distance  was  intended  to  be  one  ten-millionth  of  the 
distance  on  the  earth's  surface  from  the  equator  to  either 
pole ;  but  it  is  now  known  to  be  a  trifle  less  than  this  frac- 
tion. The  meter  is  equal  to  39.37  inches.  Its  advantage 
over  the  yard  lies  in  the  fact  that  its  subdivisions  are  deci- 
mal fractions. 

The  tenth  part  of  a  meter  is  called  a  decimeter  (dm.), 
the  hundredth  part  a  centimeter  (cm.),  and  the  thou- 
sandth part  a  millimeter  (mm.).  The  centimeter  is  the 
customary  unit  of  length  for  scientific  purposes,  and  is  the 
only  one  that  the  pupil  will  ordinarily  use  in  the  laboratory. 
Thus  a  length  of  3  dm.  5  cm.  7.5  mm.  is  written  35.75  cm. 
An  inch  is  approximately  2.5  cm.  (See  Tables  I  and  II 
of  the  Appendix.) 

The  square  centimeter  (sq.  cm.  or  cm.2)  and  the  cubic 
centimeter  (cc.,  ccm.,  or  cm.3)  are  the  customary  units  of 
area  and  of  volume  respectively.  Since  a  square  decimeter 
is  10  cm.  in  length  and  in  width,  it  contains  100  sq.  cm.; 
and  since  a  cubic  decimeter  is  10  cm.  in  each  of  its  three 
dimensions,  it  contains  1000  ccm.  A  cubic  decimeter, 
when  used  as  the  unit  of  liquid  measure,  is  called  a  liter. 
It  is  slightly  greater  than  a  quart. 


PHYSICAL  MEASUREMENTS  17 

15.  Weight.  —  The  weight  of  a  body  (Art.  9)  is  constant 
at  any  one  place  on  the  earth,  but  decreases  slightly  with 
increase  of  altitude  above  the  general  level  of  the  earth, 
as  when  a  body  is  carried  up  a  mountain  or  up  in  a  balloon. 
A  body  weighing  500  Ib.  at  sea-level  would  lose  one  pound 
of  its  weight  when  taken  to  a  height  of  about  4  miles. 
The  weight  of  a  body  increases  slightly  as  it  is  taken  from 
the  equator  toward  either  pole.     This,  as  will  be  explained 
later,  is  partly  due  to  the  rotation  of  the  earth  and  partly 
to  the  fact  that  the  earth  is  not  a  perfect  sphere.     A  body 
weighing  189  Ib.  at  either  pole  would  weigh  only  188  Ib. 
at  the  equator. 

16.  Mass.  —  The  quantity  of  matter  in  a  body  remains 
constant  unless  ,  some  portion  of  it  is  removed  or  other 
matter  added  to  it ;  but  the  volume  of  a  body  can  be  changed 
in  various  ways  without  gain  or  loss  of  matter.     For  ex- 
ample, a  fixed  quantity  of  air  or  other  gas  can  be  compressed 
to  one  half,  one  tenth,  or  one  thousandth  of  its  original 
volume ;  or  it  -can  be  allowed  to  expand  to  any  number  of 
times  its  original  volume.     So  also  100  cu.  cm.  of  ice-cold 
water  expands  to  104  cu.  cm.  when  heated  to  the  boiling 
point,  or  to  109  cu.  cm.  when  frozen;  but  there  is  no  gain 
of  matter  with  the  increase  of  volume  in  either  case.     Evi- 
dently the  volume  of  a  body  cannot  be  taken  as  the  meas- 
ure of  the  quantity  of  matter  in  it. 

On  the  other  hand,  the  weight  of  a  given  portion  of  mat- 
ter at  any  one  place  on  the  earth's  surface  remains  con- 
stant under  all  conditions.  The  weight  of  a  body  may, 
therefore,  be  taken  as  the  measure  of  the  quantity  of  mat- 
ter in  it.  Moreover,  other  facts  that  we  are  not  prepared 
to  consider  here  show  that  equal  weights  of  all  substances 
contain  equal  quantities  of  matter. 


i8  MATTER    AND    FORCE 

The  quantity  of  matter  in  a  body  is  called  its  mass.  It 
follows  from  the  above  that  the  mass  of  a  body  is  meas- 
ured by  its  weight,  and  that  any  two  bodies  having  equal 
weight  (at  the  same  place)  have  equal  mass. 

Although  the  weight  of  a  body  changes  slightly  when  it 
is  taken  to  a  different  latitude  or  a  different  altitude,  its 
mass  remains  absolutely  constant,  for  a  change  of  location 
does  not  involve  a  gain  or  a  loss  of  substance. 

17.  Units  of  Mass  and  of  Force.  —  The  principal  unit  of 
mass  in  the  metric  system  is  the  gram.  Like  the  meter, 
it  is  now  defined  with  reference  to  a  standard  kept  at  Paris. 
It  was  originally  taken  as  the  mass  of  a  cubic  centimeter 
of  distilled  water  at  the  temperature  of  its  greatest  density 
(nearly  ice-cold),  and  this  is  the  useful  definition  for  the 
purposes  of  elementary  physics.  The  mass  of  a  cubic  centi- 
meter of  fresh  water  at  any  moderate  temperature  is  so 
nearly  equal  to  one  gram  that  the  difference  may  be  dis- 
regarded. Large  masses  are  generally  expressed  in  kilo- 
grams, a  kilogram  being  equal  to  1000  grams.  The  pound 
mass  is  the  principal  unit  of  mass  in  the  English  system. 

The  weight  of  a  unit  mass  is  taken  as  a  unit  of  force. 
The  familiar  unit,  of  course,  is  the  pound  weight.  Thus  if 
we  say  that  a  horse  exerts  a  pull  of  150  Ibs.  in  drawing 
a  load,  we  mean  that  the  pull  is  equal  to  the  earth's  attrac- 
tion for  a  i5o-pound  mass.  The  weight  of  a  gram  mass 
is  the  principal  metric  unit  of  force.  A  unit  of  mass  and 
the  corresponding  unit  of  force  have  the  same  name.  Thus 
we  speak  of  a  mass  of  so  many  grams,  meaning  a  certain 
quantity  of  matter,  or  a  force  of  so  many  grams,  meaning 
a  certain  push  or  pull.  This  double  use  of  the  terms  is 
unfortunate;  but  one  can  always  tell  from  the  connec- 
tion in  which  they  are  used  whether  mass  or  force  is 
referred  to. 


PHYSICAL  MEASUREMENTS  19 

From  what  has  been  said  concerning  the  variation  of 
weight,  it  is  evident  that  a  unit  of  weight  is  not  exactly  the 
same  at  all  places  on  the  earth's  surface;  but  the  variation 
is  so  slight  as  not  to  be  a  matter  of  practical  importance. 

18.  The  Measurement  of  Mass  (Weighing).  —  We  make 
use  of  the  equal  attraction  of  the  earth  for  equal  masses 
in  weighing  with   an   equal-arm  balance  (Fig.   4).     The 
arms  are  the  two  halves  of  the  beam,  from  which  the 
pans    are    suspended.      When 

equal  downward  forces  are  ex- 
erted on  the  pans,  the  beam 
comes  to  rest  in  a  horizontal 
position.  Hence  if  the  beam 
takes  this  position  when  there 
is  a  certain  load  in  each  pan, 
we  know  that  the  loads  have 
equal  weight  and  consequently 
equal  mass.  FIG.  4. 

To  find  the  mass  of  a  body  it  is  placed  in  one  pan 
and  balanced  with  standard  masses  in  the  other.  The 
process  is  called  weighing,  and  the  standard  masses  are 
commonly  called  weights.  (Note  that  in  this  sense  a 
"weight"  is  a  certain  standard  piece  of  matter,  not  a 
force.) 

19.  The  Unit  of  Time.  —  The  rotation^  of  the  earth  on 
its  axis  is  constant.     The  period  of  one .  complete  rota- 
tion is,  therefore,  an  invariable  natural  unit  of  time,  and 
is  called  a  "sidereal  day."     Even  the  best  chronometers 
are  not  perfectly  accurate,  and  they  are  corrected  by  com- 
parison with  the  earth's  rotation,  as  determined  by  the 
apparent  motion  of  the  stars.     Owing  to  the  earth's  annual 
motion  round  the  sun,  the  solar  day,  or  the  time  from  "high 


20  MATTER    AND    FORCE 

noon"  to  "high  noon,"  is  slightly  longer  than  the  sidereal 
day  and  is  also  slightly  variable.  The  average  length  of 
the  solar  day  for  the  entire  year,  or  the  "mean  solar  day," 
is  divided  into  24  hours,  the  hour  into  60  minutes,  and  the 
minute  into  60  seconds.  These  are  the  time  intervals 
indicated  by  clocks  and  watches.  The  second  is  the  unit 
of  time  regularly  used  in  scientific  work. 

20.  Fundamental  Units.     English  and  Metric  Systems. 
-  We  have  seen  that  the  units  of  surface  and  of  volume 

are  derived  from  the  units  of  length.  Velocity  is  expressed 
in  terms  of  a  unit  of  length  and  a  unit  of  time,  e.g.  in  miles 
per  hour,  feet  per  second,  etc.  Similarly,  almost  all  phys- 
ical quantities  (and  there  are  many)  can  be  expressed  in 
terms  of  the  units  of  length,  mass,  and  time;  hence  these 
are  called  the  fundamental  units.  The  fundamental  units 
of  the  metric  system  are  the  centimeter,  the  gram  mass, 
and  the  second;  from  which  it  is  often  termed  the  centi- 
meter-gram-second or  the  C.G.S.  system.  The  foot,  the 
pound  mass,  and  the  second  are  the  fundamental  units 
of  the  English  or  foot-pound-second  (F.P.S.)  system. 

21.  Density. — A  piece   of  iron  weighs  more   (has   a 
greater  mass)  than  a  piece  of  wood  of  the  same  size.     In 
ordinary  language  we  say  that  iron  is  heavier  than  wood, 
wood  is  lighter  than  water,  cork  is  very  light,  etc.     It  is 
understood  that  such  statements  refer  to  the  weights  of 
equal  volumes  of  the  substances;  but  the  language  is  not 
exact,  and  hence  is  not  adapted  to  scientific  use. 

The  mass  of  a  unit  volume  of  a  substance  is  called  its 
density.  The  density  of  pure  cold  water  is  i  gram  per 
cu.  cm.  (by  definition  of  the  gram  mass),  or  62.4  pounds 
per  cu.  ft.  The  density  of  cast  iron  is  7.2  g.  per  cu.  cm.  or 


x      PHYSICAL  MEASUREMENTS  21 

449  Ib.  per  cu.  ft.     Its  density  is  thus  7.2  times  as  great  as 
the  density  of  water. 

The  density  of  a  substance  is  determined  by  measuring 
the  mass  and  the  volume  of  any  convenient  portion  of  it, 
and  computing  from  these  measurements  the  mass  of  one 
cubic  centimeter. 

PROBLEMS 

1.  Would  the  weight  of  a  body  appear  to  differ  in  different  latitudes  and 
at  different  altitudes;    (a)  when  accurately  determined  with  an  equal-arm 
balance;   (b)  when  accurately  determined  with  a  spring  balance?     Give  the 
reasons  for  your  answers. 

2.  Is  the  density  of  a  body  affected  by  taking  it  to  a  different  latitude 
or  altitude? 

3.  Is  abound  of  iron  heavier  than  a  pound  of  wood?    What  is  implied 
in  the  familiar  statement  that  "iron  is  heavier  than  wood"?     Show  that  the 
statement  that  "iron  is  denser  than  wood"  leaves  nothing  to  be  implied. 
Which  form  of  statement  is  to  be  preferred? 

4.  Letting  v  denote  the  volume  of  a  body,  d  its  density,  and  m  its  mass, 
write  the  formula  (equation)  expressing  the  relation  of  these  three  quantities 
to  one  another.    Write  this  equation  expressing  (a)  the  value  of  m  in  terms 
of  v  and  d;  (b)  the  value  of  d  in  terms  of  v  and  m;  (c)  the  value  of  v  in 
terms  of  d  and  m. 

Note.  —  In  physics  it  is  customary  to  represent  the  value  of  a  physical 
quantity,  whether  known  or  unknown,  by  the  initial  letter  of  its  name. 

5.  The  volume  of  a  stone  is  630  cm.;    its  mass  is  1575  g.     Find  its 
density. 

6.  What  is  the  volume  of  1000  g.  of  mercury?  Of  1000  g.  of  brass?  Of 
1000  g.  of  aluminum?     (See  table  of  densities  in  the  Appendix.) 

*7.   What  is  the  mass  of  i  cu.  dm.  of  lead?  Of  i  cu.  dm.  of  marble? 

8.  Find  the  densities  of  water,  quartz,  and  gold  in  pounds  per  cubic 
foot,  from  the  densities  in  grams  per  cubic  centimeter  given  in  the  table. 
(See  also  the  table  of  equivalents  in  the  Appendix.) 

9.  From  the  known  densities  of  ice  and  water,  show  whether  water 
expands  or  contracts  in  freezing. 

10.  Criticize  the  statement:    i  ccm.  =  i  g.;  also  the  statement  i  ccm. 
of  water  =  i  g.     In  what  different  ways  may  the  truth  of  the  matter  be 
correctly  expressed? 


CHAPTER  III 

STATICS  OF  LIQUIDS 

I.     INTRODUCTION 

22.  The  Problems  of  Mechanics  may  be  briefly  described 
as  problems  in  equilibrium  and  problems  in  motion.     On 
the  basis  of  this  classification  Mechanics  is  subdivided 
into  Statics  and  Dynamics.     The  present  chapter  deals 
with  the  equilibrium  of  liquids,  or  Hydrostatics. 

Common  observation  teaches  that  liquids  exert  pressures.  Pipes, 
tanks,  and  dams  must  have  a  certain  strength  to  resist  the  pressure 
of  water,  or  they  will  burst.  A  ship  rides  safely  on  the  water;  yet 
its  enormous  weight  has  no  other  support  than  the  yielding  liquid. 
The  designer  of  a  ship  must  know  before  the  keel  is  laid  how  far  it 
will  sink  when  launched,  and  how  much  farther  with  a  full  cargo. 
He  must  also  know  that  the  ship  will  float  upright,  and  not  "  turn 
turtle." 

Evidently  the  mechanical  behavior  of  liquids  and  of 
bodies  floating  on  them  or  immersed  in  them  is  definite  and 
dependable.  To  understand  this  behavior  one  must  know 
the  general  facts  or  principles  of  liquid  pressure.  A  few 
preliminary  ideas  concerning  pressure  in  solids  will 
help  us. 

23.  Transmission  of  Pressure  Through  Solids.     Applied 
Pressure  and  Gravity  Pressure.  —  When  an  object  is  pushed 
with  a  stick  held  in  the  hand,  the  force  (pressure)  is  trans- 
mitted from  the  hand  to  the  object  through  the  stick.     The 
stick  sustains  the  pressure  throughout  its  length;  and,  if 

22 


INTRODUCTION  23 

it  is  not  strong  enough  to  withstand  this  pressure,  it  will 
bend  or  break  at  the  weakest  place.  This  is  an  example 
of  an  applied  pressure  acting  upon  and  transmitted  through 
a  solid.  The  weight  of  any  body  gives  rise  to  a  pressure 
which  is  similarly  transmitted  throughout  the  body.  In  a 
brick  wall,  for  example,  each  brick  transmits  to  those  be- 
neath it  the  pressure  exerted  upon  it  by  all  the  overlying 
bricks,  and  adds  to  that  a  pressure  equal,  to  its  own  weight. 
The  pressure  therefore  increases  from  top  to  bottom  of 
the  wall,  and  at  any  level  its  amount  is  determined  by  the 
weight  of  the  overlying  bricks.  Such  pressures  are  called 
gravity  or  weight  pressures,  to  distinguish  them  from  exter- 
nal or  applied  pressures.  The  pressure  at  the  bottom  of  a  fac- 
tory chimney  is  a  gravity  pressure  when  considered  with 
respect  to  the  chimney  in  which  the  pressure  originates; 
it  is  an  applied  pressure  when  considered  as  an  external 
force  acting  on  the  foundation  which  supports  the  chimney. 

24.  Lateral  Pressure  Due  to  Weight.  — The  gravity  pres- 
sure in  a  wall  of  masonry  is  vertical.  Each  brick  or  stone 
presses  up  on  its  neighbors  above  and  down  on  its  neigh- 
bors below,  but  not  laterally  on  its  neighbors  at  the  same 
level.  In  a  pile  of  sand  or  shot  each  individual  crowds 
in  between  its  neighbors,  causing  a  pressure  sideways  as 
well  as  upward  and  downward,  as  is  shown  by  the  tendency 
of  the  pile  to  spread  at  the  bottom.  To  make  the  sides  of 
the  pile  vertical,  supporting  surfaces  must  be  provided  to 
sustain  the  lateral  pressure. 

Similarly,  the  weight  of  a  liquid  causes  lateral  and  oblique 
as  well  as  vertical  pressures  within  it.  These  pressures 
are  more  fully  developed  in  liquids  than  in  a  pile  of  shot, 
for  the  particles  of  a  liquid  are  free  to  move  over  one  an- 
other, while  in  shot  there  is  considerable  friction.  Hence 
shot  remains  in  a  sloping  pile  and  a  liquid  does  not. 


STATICS    OF    LIQUIDS 


\ 


u 


\ 

FIG.  5. 


II.     GRAVITY  PRESSURE  IN  LIQUIDS 

25.  Relation  Between  Pressure  and  Depth.  Pressure 
in  Different  Directions.  —  The  pressure  at  different  depths 
and  in  different  directions  in  water  can  be 
observed  with  the  aid  of  glass  tubes  of  equal 
length  (60  cm.  or  more),  closed  at  the  top 
and  shaped  at  the  lower  end  as  shown  in 
Fig.  5.  When  such  a  tube  is  lowered  into 
a  tall  glass  jar  filled  with  water,  the  water 
enters  its  lower  end  to  a  greater  or  less  dis- 
tance according  to  the  pressure ;  for  the  water 
can  enter  only  as  the  confined  air  is  com- 
pressed, and  the  compression  increases  with  the  pressure. 
The  water  pushes  farther  in  as  the  tube  is  lowered,  showing 
that jthe_pressuie,  increases  with-the  depth.  When  the  differ- 
ent tubes  are  inserted  to  the  same  depth,  the  water  enters 
an  equal  distance  in  all,  showing  that  the  pressure  at  a  given 

depth  is  the   same   in   the 
various  directions  tested. 

By  other  methods  which 
permit  exact  measurement 
it  is  found  that  the  gravity 
pressure  at  any  poinl  in  a  liquid  at 
rest  is  proportional_Jo_Jhe  depth  of 
the  point  below  the  free  surface  of 
the  liquid,  and  at  any  point  the  pres- 
sure is  the  same  in  all  directions. 

This  behavior  is  explained  as 
follows.    The  pressure  at  any  level 
is  due  to  the  weight  of  the  over- 
lying liquid; 'and,  since  the  liquid 
FlG.  6.  is  of  the  same  density  at  all  depths 


GRAVITY  PRESSURE  IN  LIQUIDS  25 

(liquids  being  practically  incompressible),  the  weight  of 
liquid  in  a  vertical  column  is  proportional  to  the  depth  of 
the  column.  Thus  at  a  depth  of  2  cm.  the  pressure  is  twice 
as  great  as  at  a  depth  of  i  cm. ;  at  3  cm.  it  is  three  times 
as  great,  etc.  (Fig.  6).  Further,  since  each  particle  of 
the  liquid  is  free  to  move,  it  would  not  remain  at  rest  if  the 
pressures  upon  it  in  different  directions  were  unequal. 

26.  Direction  of  Fluid  Pressure  Against  a  Surface. — The 

pressure  of  a  liquid  at  rest  is  perpendicular  to  any  surface 
against  which  it  is  exerted,  e.g.  the  walls  of  the  containing 
vessel  or  the  surface  of  an  immersed  body.  This  also  is 
due  to  the  fact  that  the  particles  of  a  liquid  are  free  to  move. 
If  the  pressure  were  oblique  to  any  surface,  the  liquid 
would  flow  along  it. 

27.  Pressure  in  Vessels  of  Different  Sizes  and  Shape.  — 

Let  a  funnel  and  a  glass  tube  be  connected  by  a  rubber 
tube  and  partly  filled  with  water  (Fig.  7).  The  water 
stands  at  the  same  level 
in  the  funnel  and  the 
tube,  whether  they  are 
vertical  or  inclined  at 
any  angle.  This  beha- 
vior Suggests  One  Of  the  FlG-  7-  —  A  liquid  "seeks  its  own  level." 

most  important  general  truths  in  the  mechanics  of  liquids. 
Imagine  a  plane  cutting  across  the  tube  at  its  lowest  part 
m.  The  pressure  'of  the  water  in  the  funnel  tends  .to 
push  water  past  m  and  up  into  the  glass  tube\,  while 
the  pressure  of  the  water  in  the  glass  tube  tends  to  push 
water  past  m  in  the  opposite  directionand  up  into  the  fun- 
nel. Since  there  is  no  flow  in  either  direction,  these  grav- 
ity pressures  must  be  equal.  When  either  side  is  lowered 
or  inclined,  equilibrium  is  destroyed,  and  water  immedi- 


26 


STATICS    OF    LIQUIDS 


ately  flows  to  that  side  until  the  two  surfaces  are  again  at 
the  same  level.  Evidently  the  pressures  are  equal  at  the 
bottom  of  the  funnel  and  the  tube  only  when  the  depth 
of  the  water  (measured  vertically)  is  the  same  in  both. 
The  greater  mass  (or  weight)  of  the  water  in  the  funnel 
does  not  affect  the  result  in  the  least.  This  agrees  with 
the  well-known  fact  that  water  stands  at  the  same  height 
in  the  spout  of  a  kettle  as  in  the  body  of  the  vessel,  or  the 
fact  that  water  will  rise  in  a  pipe  only  to  the  level  of  the 
surface  in  the  tank  or  reservoir  from  which  it  comes.  In 
general— 

The  gravity  pressure  of  a  liquid  at  a  given  depth,  either 
within  the  body  of  the  liquid  or  against  any  surface,  is  wholly 
independent  of  the  size  and  shape  of  the  vessel. 

This  can  be  further  shown  with  three  vessels,  a,  b,  and  c  (Fig.  8), 
having  bottoms  of  the  same  area  and  filled  to  the  same  depth  with  the 
same  liquid.  A  disk,  A,  serves  as  a  bottom  for  each  of  the  vessels 


FIG.  8. — The  factors  of  gravity  pressure  are  depth  and  density  only. 

in  turn,  being  held  in  place  by  the  upward  pull  of  a  cord.  This  cord 
is  attached  to  an  arm  of  a  balance,  by  means  of  which  the  same  pull 
is  exerted  in  each  case.  With  the  adjustment  shown  for  vessel  a, 
the  liquid  is  poured  in  till  the  pressure  becomes  great  enough  to  force 
the  disk  from  the  bottom.  It  will  be  found  that  this  requires  the 
same  depth  of  liquid  in  the  three  vessels.  Only  with  the  vessel  a, 


GRAVITY    PRESSURE  IN  LIQUIDS 


27 


however,  is  the  downward  force  of  the  liquid  on  the  disk  equal  to  the 
weight  of  the  liquid.  The  vertical  sides  of  this  vessel  do  not  help  to 
support  the  weight  of  the  liquid.  In  vessel  b  the  liquid  is  partly 
supported  by  the  slanting  sides,  which  press  obliquely  upward  against 
the  liquid  (the  pressure  being  perpendicular  to  the  surface).  In 
vessel  c  the  slanting  part  of  the  side  presses  obliquely  downward 
against  the  liquid,  thus,  in  effect,  supplying  a  part  of  the  downward 
force  which  is  exerted  on  the  bottom. 

28.  Relation  Between  the  Pressure  of  a  Liquid  and  its 
Density.  —  We  have  seen  that  the  gravity  pressure  of  a 
liquid  against  any  surface  is  proportional  to  the  depth  of 
the  liquid  above  it,  because  the  weight  of  the  vertical  col- 
umn of  liquid  overlying  the  surface  is  proportional  to  the 
depth.  But  the  weight  of  such  a  column  is  also  propor- 
tional to  the  density  of  the  liquid;  hence 
we  should  expect  the  gravity  pressure  at 
equal  depths  in  different  liquids  to  be  pro- 
portional to  their  densities.  Experiment 
shows  that  this  is  the  case. 

If  a  small  quantity  of  mercury  is  poured  into  a 
tall  U-tube,  and  one  of  the  arms  is  then  nearly 
filled  with  water,  the  liquids  will  stand  as  shown 
in  Fig.  9.  The  pressure  at  the  same  level  c  and 
d  in  the  two  arms  must  be  equal,  since  below  :  _ff 

that  level  we  have  the  same  liquid  on  both  sides. 
But  the  pressure  at  c  is  due  to  the  column  of  water 
be,  and  the  pressure  at  d  to  the  mercury  column  ad. 
Now  it  is  found  by  measurement  that  the  water     jt'y  pressures  at 
column  is  13.6  times  as  high  as  the  mercury  column     c    and     d    are 
ad.     Since  therefore  a  column  of  mercury  rs-s  as     equa ' 
high  as  the  water  column  exerts  an  equal  pressure,  a  mercury  col- 
umn of  the  same  height  as  the  water  would  exert  a  pressure  13.6 
times  as  great.     But  the  density  of  mercury  is  13.6  times  that  of 
water;  which  agrees  with    the  general  conclusion  that  the  gravity 
pressures  at  equal  depths  in  different  liquids  are  proportional  to  the 
densities  of  the  liquids. 


28  STATICS    OF    LIQUIDS 

29.  Summary    of    the    Laws    of    Gravity  Pressure    in 
Liquids. — The  general  facts  or  laws  of  liquid  pressure 
which  we  have  been  considering  are  as  follows: 

1.  'The  pressure  at  any  point  in  a  fluid  at  rest  is  the  same 
in  all  directions. 

2.  The  pressure  of  a  fluid  at  rest  is  perpendicular  to  any 
surface  with  which  it  is  in  contact. 

3.  The  gravity  pressure  in  a  liquid  at  rest  is  proportional 
to  the  depth  and  to  the  density  of  the  liquid. 

The  first  two  laws  hold  for  gases  as  well  as  for  liquids 
and  for  both  gravity  pressure  and  applied  pressure;  hence 
the  more  general  form  in  which  they  are  stated. 

30.  Digression  on  Natural  Laws.  —  Experience  teaches 
that,  in  nature,  wherever  and  whenever  the  same  conditions 
are  repeated,  the  same  results  follow.     Natural  phenomena, 
when  fully  understood,  always  disclose  uniformity,  order, 
system.     Without  this  uniformity  in  nature,  science  would 
be  impossible,  and  the  innumerable  applications  of  scien- 
tific principles  which  we  see  on  every  hand  would  also  be 
impossible.     To  mention  a  single  instance,  if  the  behavior 
of  electricity  under  the  same  conditions  were  not  invariable 
and  dependable,  electrical  power  could  never  have  been 
brought  under  control  and  made  the  tremendously  useful 
servant  that  it  is  at  the  present  day. 

The  uniform  behavior  of  matter,  or  the  unvarying  course 
of  phenomena,  under  the  same  conditions  is  known  as  a 
natural  law.  A  natural  law  is  a  fact  in  nature  before  it  is 
discovered,  as  well  as  afterward.  The  laws  of  gravity  pres- 
sure in  liquids  and  the  law  of  applied  pressure  (Art.  34) 
were  discovered  by  the  French  scientist  Blaise  Pascal 
about  the  middle  of  the  seventeenth  century.  Laws  of 
nature  within  the  domain  of  physics  are  called  physical 
laws.  They  are  met  with  in  considerable  number  in  all 


GRAVITY  PRESSURE  IN  LIQUIDS  29 

branches  of  the  subject.  Thus  in  certain  respects  all  gases 
behave  alike,  and  this  uniform  behavior  constitutes  the 
laws  of  gases.  In  certain  respects  all  bodies  behave  alike 
under  the  action  of  force,  and  this  uniform  behavior  con- 
stitutes the  laws  of  motion.  Light  is  always  reflected  in 
a  definite  manner  from  polished  surfaces,  and  this  beha- 
vior is  known  as  the  law  of  reflection  of  light. 

When  a  natural  law  is  discovered,  the  dicoverer  formu- 
lates a  statement  of  the  fact  as  he  sees  it.  This  statement 
is  itself  called  a  law  (sometimes  a  principle).  Laws  in 
this  sense  are  brief  descriptive  statements  of  the  manner  of 
behaving  —  not  the  behavior  itself.  It  is  well  to  bear  in 
mind  these  two  meanings  of  the  word  law,  as  used  in  sci- 
ence. Natural  processes  are  never  amended;  but  state- 
ments of  what  men  believe  to  be  the  fact  sometimes  require 
modification  in  consequence  of  later  and  more  accurate 
or  more  complete  information  on  the  subject. 

31.  Pressure  as  a  Measured  Quantity. — The  term  pres- 
sure, as  we  have  used  it  thus  far,  may  be  taken  to  mean 
either  the  whole  force  exerted  by  the  liquid  upon  any  given 
area  or  the  force  exerted  upon  each  unit  area  of  the  surface. 
Whenever  we  have  to  deal  with  fluid  pressures  numerically, 
it  is  necessary  Jx>  restrict  the  use  of  the  term  to  one  of 
these  two  possible  meanings,  in  order  that  there  may  be 
no  misunderstanding;  and  it  is  agreed,  among  both  scien- 
tists and  engineers,  that  pressure  shall  always  mean  the 
force  per  unit  area.  Thus  when  an  engineer  speaks  of  a 
boiler  pressure  of  150  Ib.  he  means  that  the  steam  exerts 
a  force  of  150  Ib.  against  each  square  inch  of  the  boiler 
surface.  Pressure  is  measured  in  pounds  per  square  inch, 
pounds  per  square  foot,  grams  per  square  centimeter,  etc. 

The  force  exerted  by  a  fluid  against  the  whole  of  a  given 
area  (whether  greater  or  less  than  a  unit  area)  is  called 


30  STATICS  OF  LIQUIDS 

the  total  force  or  the  thrust.  This  distinction  of  terms  is 
strictly  adhered  to  in  all  that  follows,  both  in  the  mechanics 
of  liquids  and  the  mechanics  of  gases.  The  term  pressure 
at  a  point  means  the  force  per  unit  area  at  the  level  of  the 
point. 

The  rules  for  computing  gravity  pressure  are  derived 
from  the  laws.  The  thrust  of  a  liquid  against  the  bottom 
of  a  vessel  having  vertical  sides  is  equal  to  the  weight  of 
the  liquid.  (Why?)  The  pressure  on  the  bottom  is  equal 
to  the  weight  of  the  vertical  column  of  liquid  whose  base 
is  any  unit  area  of  the  bottom,  and  this  is  equal  to 
the  product  of  the  depth  and  the  density  of  the  liquid. 
(Why?)  Hence  the  general  rule: 

The  gravity  pressure  at  any  point  in  a  liquid  is  equal  to 
the  product  of  the  depth  of  the  point  below  the  free  surface 
of  the  liquid  and  the  density  of  the  liquid. 

Thus  in  either  of 
the  vessels  shown  in 
Figs.  10  and  n,  the 
pressure  at  the  level 
MN  is  everywhere 
the  same,  and  is 
equal  to  hd  g.  per 
sq.  cm.,  where  h  rep- 
resents the  depth  of 
MN  below  the  free  surface  of  the  liquid  and  d  the  density  of 
the  liquid. 

32.  Rules  for  Computing  the  Thrust.  —  Horizontal  Sur- 
faces. —  Since  the  pressure  of  a  liquid  is  uniform  over  a 
horizontal  surface,  the  thrust  against  such  a  surface  is 
equal  to  the  product  of  the  pressure  and  the  area. 

Oblique  and  Vertical  Surfaces.  —  The  pressure  against  a 
vertical  or  an  oblique  plane  surface  increases  uniformly 
from  the  upper  to  the  lower  side.  To  find  the  thrust 


FlG.   10. 


FIG.  ii. 


GRAVITY  PRESSURE  IN  LIQUIDS 


31 


against  such  a  surface,  the  average  pressure  upon   it  is 
multiplied  by  the  area.     This  average  pressure  is  equal  to  , 
the  actual  pressure  at  the  center  of  the  surface. 

For  example,  the  thrust  against  a  water  gate  which 
is  6  ft.  wide  and  4  ft.  high,  and  the  top  of  which  is  8 
ft.  below  the  surface,  is  (10  X  62.4)  X  (4  X  6)  =  14,976  Ib. 

33.  Applications  of  Gravity  Pressure.  —  Gravity  pres- 
sure is  utilized  in  the  water-supply  of  cities.  Wherever 
possible,  reservoirs  are  located  at  a  sufficient 
elevation  to  provide  the  necessary  pressure. 
Distributing  pipes  or  mains  carry  the  water  to 
all  parts  of  the  city,  and  these  connect  with  the 
water-pipes  in  each  house.  When  much  water 
is  flowing  in  the  pipes  there  is  considerable 
loss  of  pressure  due  to  friction,  and  water  will 
not  rise  in  the  pipes  to  the  level  of  its  source. 
The  reservoir  must  therefore  be  higher  than 
any  point  to  be  supplied  from  it.  When  the 
reservoir  can  not  be  located  at  a  sufficient 
height  to  supply  the  distributing  system  directly, 
the  necessary  pressure  is  maintained  by  pumping 
the  water  from  the  reservoir  (or  sometimes  from 
a  lake)  into  an  elevated  tank  or  a  tall  stand-pipe 
(Fig.  12). 


FIG.  12.  — This  stand- 
pipe  at  Erie,  Pa., 
holds  the  water  235 
ft.  above  the  level  of 
Lake  Erie,  whence 
the  supply  is  drawn; 
e,  the  pump  house; 
w,  the  top  of  the  col- 
umn of  water  in  the 
stand-pipe. 


FIG.  13. —  Artesian  Wells. 

The  flow  of  artesian  wells  is  due  to  the  gravity  pressure  of  water. 
If  a  porous  stratum  of  sand  or  gravel  (c,  Fig.  13),  lying  between  two 
impervious  strata  and  dipping  under  a  lower  flat  country,  becomes 
filled  with  water  above  the  level  of  the  ground  where  a  well  is  bored, 
an  artesian  or  flowing  well  will  result  (Fig.  14). 


32 


STATICS  OF  LIQUIDS 


The  use  of  water-power  for  running  turbine  water-wheels.  (Fig. 
159  6)  is  an  important  application  of  gravity  pressure.  The  wheel 
is  at  the  bottom  of  a  large  pipe,  which  conducts  the  water  to  it 

from  a  higher  level,  under  a  pres- 
sure which  depends  upon  the  ele- 
vation or  "  head  "  of  the  water  above 
the  wheel.  The  wheels  of  the  Niag- 
ara Falls  Power  Company  are  sup- 
plied with  water  under  a  head  of 
136  feet. 

PROBLEMS 

1.  What  is  the  gravity  pressure  (a) 
at  a  depth  of    20  cm.  in  water?  (6)  at 
a  depth  of  60  cm.  in  mercury?  (c)  at  a 
depth  of  50  cm.  in  alcohol?     (See  table 
of  densities  in  the  Appendix.) 

2.  What  is  the  pressure  in  pounds 
per  square  foot  (a)  at  a  depth  of  20  ft.  in 
water?    (6)  at  a  depth  of  3  mi.  in  the 
ocean?     (Take  62.4  Ib.  per  cu.  ft.  as  the 
density  of  pure  water  in  all  problems. 
The  density  of  sea  water  is  1.026  times 
this.) 

3.  A  rectangular  vessel  50  cm.  long, 
20  cm.  high,  and  35  cm.  wide   is  filled 

liquid  whose   density  is    1.5    g.  per  ccm.     Find  the  thrust  (a) 
against  the  bottom;  (b)  against  a  long  side. 

4.  Find  the  thrust  in  pounds  against  the  side  of  a  cylindrical  tank,  15 
ft.  in  diameter  and  12  ft.  high,  when  filled  with  water. 

5.  Find  the  thrust  against  a  vertical  dam,  100  ft.  long,  against  which 
water  stands  to  a  depth  of  12  ft. 

6.  A  round  hole  2  in.  in  diameter,  in  the  side  of  a  water  tank,  is  closed 
with  a  plug.      What   is  the   thrust  against   the   plug   when  the   water 
stands  8  ft.  deep  above  the  center  of  the  hole  ? 

7.  What  is  the  pressure  in  pounds  per  square  inch  due  to  a  90  ft.  head 
of  water? 

8.  An  outlet  through  the  side  of  a  dam  is  closed  by  a  gate  4  ft.  wide  and 
3  ft.  high;   and  the  top  of  the  gate  is  9  ft.  below  the  surface  of  the  water. 
What  thrust  does  the  gate  sustain? 


FIG.  14. —  Artesian  Well  at  Woon- 
socket,  S.  D.  When  photo- 
graphed, the  jet  was  97  ft.  high. 


with 


FlG- 


TRANSMISSION  OF  APPLIED  PRESSURE  33 

III.   TRANSMISSION  OF  APPLIED  PRESSURE  BY  LIQUIDS 

34.  Pascal's  Law.  —  A  liquid  transmits  pressure  in  the 
same  way  whether  the  pressure  is  due  to  its  own  weight  or 
to  an  applied  force.  For  example,  suppose  a  vessel,  A 
(Fig.  15),  to  be  filled  with  water 
to  the  level  MN,  the  depth  c 
being  10  cm.  The  gravity  pres- 
sure of  the  water  against  the 
bottom  will  be  10  g.  per  square 
centimeter.  Now  if  water  is 
added  to  the  additional  depth  of 
20  cm.,  or  to  the  level  CD,  the 
pressure  on  the  bottom  will  be  30  g.  per  square  centime- 
ter, and  everywhere  throughout  the  original  body  of  water 
the  pressure  will  be  20  g.  per  square  centimeter  greater 
than  it  was  before.  If  this  additional  pressure  were  exerted 
by  means  of  a  tight-fitting  piston,  as  shown  in  B,  it 
would  be  regarded  as  an  external  or  applied  pressure;  but 
its  effect  would  be  the  same  as  in  the  first  case,  i.e.  the 
applied  pressure  of  20  g.  per  square  centimeter  would  be 
transmitted  throughout  the  water,  adding  just  that  much 
to  the  original  pressure  against  every  unit  area  of  the  bot- 
tom and  sides  of  the  vessel.  In  general  — 

A  pressure  applied  to  any  part  of  an  inclosed  fluid  is 
transmitted  throughout  the  fluid,  with  unchanged  intensity,  to 
all  parts  of  the  interior  surface  of  the  vessel,  and  its  direction 
is  everywhere  perpendicular  to  the  surfqce.  This  is  known  as 
Pascal's  law,  after  the  French  mathematician  and  physi- 
cist, Blaise  Pasca^  by  whom  it  was_discovejred. 

An  important  consequence  of  Pascal's  law  is  illustrated  in  Fig. 
1 6.  The  apparatus  consists  of  two  cylinders  having  unequal  diame- 
ters, connected  together  and  fitted  with  pistons.  The  pistons  rest 


34 


STATICS  OF  LIQUIDS 


upon  the  water  in  the  vessel,  and  a  pressure  exerted  by  either  is 
transmitted  by  the  water  to  the  other.     Since  the  pressure  (force 

per  unit  area)  is  the  same  against 
both  pistons,  the  thrusts  against  them 
are  proportional  to  their  areas.  Thus 
if  the  area  of  the  larger  piston  is  50 
times  that  of  the  smaller,  a  weight  of 
i  kg.  placed  on  the  smaller  will  bal- 
ance a  weight  of  50  kg.  on  the  larger. 
It  will  be  seen  from  this  that  water 
(or  any  other  fluid)  is  an  effective 
instrument  for  transmitting  and  multiplying  force.  This  mechanical 
principle  is  applied  in  a  great  variety  of  hydraulic  and  pneumatic 
(compressed-air)  machines  and  devices. 

35.  Applications  of  Fluid  Pressure.  —  The  hydraulic  or  hydro- 
static press  (Fig.  17,  a  and  b)  is  an  important  application  of  Pascal's 
principle.  C  is  a  very  strong 
metal  cylinder.  In  it  there  is 
a  cast-iron  piston,  P',  working 
water  tight  in  the  head  of  the 


FIG.  1 6.  —  Thrust  is  Proportional 
to  Area. 


FIG. 


170.  —  Section  Diagram  of  the 
Hydraulic  Press. 


FIG.  176. — One-thousand  Ton  Press  used 
in  Steel  Car  Construction.  The  fixed 
platform  is  at  the  bottom  of  the  press; 
the  pressure  is  applied  downward. 


cylinder.     The  top  of  the  piston  carries  an  iron  plate  M,  on  which  is 
placed  the  substance  to  be  pressed.    The  fixed  upper  plate,  N,  is 


ATMOSPHERIC    PRESSURE  51 

are  based  on  other  sources  of  information  as  well,  includ- 
ing temperature,  direction,  and  velocity  of  the  wind,  the 
course  and  progress  of  storms  up  to  the  time  when  the  fore- 
cast is  made,  and  the  existing  state  of  the  weather;  all  of 
which  are  reported  to  the  central  office  at  Washington  by 
the  different  stations  distributed  over  the  country.  The 
barometer  is  not  to  be  held  responsible  for  erroneous  fore- 
casts. Its  function  is  to  measure  atmospheric  pressure, 
and  this  it  dt>es  correctly. 

Since  the  atmospheric  pressure  changes  at  a  known  rate 
with  change  of  altitude,  the  barometer  can  be  used  for  meas- 
uring altitudes.  To  find  the  height  of  a  mountain  by  this 
method,  the  barometric  pressure  at  its  base  and  at  its  sum- 
mit are  taken  as  nearly  at  the  same  time  as  possible.  The 
height  to  which  a  balloon  ascends  is  determined  in  the  same 
way.  For  moderate  altitudes  above  sea-level,  it  is  approxi- 
mately correct  to  compute  the  change  of  altitude  at  the 
rate  of  900  ft.  for  a  fall  of  the  barometer  of  one  inch. 

PROBLEMS 

1.  Explain  the  process  of  drinking  through  a  straw. 

2.  When  the  mercury  barometer  stands  at  a  height  of  76  cm.,  what  will 
be  the  height  of  a  barometer  the  liquid  in  which  has  a  specific  gravity  of  1.6? 

3.  When  the  barometer  stands  at  76  cm.,  a  liter  of  air  at  o°  C.  weighs 
1.293  g-     At  the  same  temperature  and  pressure,  what  will  be  the  weight  of 
the  air  in  a  room  9  m.  by  7  m.  and  4  m.  high? 

4.  Compute  the  weight  of  i  cu.  ft.  of  air  at  o°  C.  and  76  cm.  pressure  (sp. 
gr.  of  air  =  .001293). 

6.  What  weight  of  air  at  this  temperature  and  pressure  is  contained  in  a 
room  20  by  30  ft.,  and  12  ft.  high? 

6.  The  force  required  to  separate  Guericke's  hemispheres,  which  were 
1.2  ft.  in  diameter,  is  equal  to  the  total  force  of  the  atmosphere  on  a  flat 
circular  area  of  the  same  diameter.  Compute  it,  assuming  a  pressure  of 
14  7  lb.  per  sq.  in. 


52  STATICS  OF  GASES 

7.  (a)  The  weight  of  the  atmosphere  is  equal  to  the  weight  of  an  ocean 
of  mercury  covering  the  entire  surface  of  the  earth  to  what  depth?  (6)  What 
would  be  the  depth  of  water  covering  the  entire  surface  of  the  earth  and 
having  equal  weight? 

II.  LAWS  OF  GASES 

50.   The  Behavior  of  Liquids  and  Gases  Compared.  — 

Liquids  and  gases  behave  alike  in  part,  since  both  are  fluids. 
The  first  and  second  laws  of  Article  29  and  Pascal's  law 
(Art.  34)  are  laws  of  fluid  pressure,  including  both  liquids 
and  gases.  But,  owing  to  the  small  density  and  great 
compressibility  of  gases',  their  behavior  in  other  respects 
is  very  different  from  that  of  liquids.  Vessels  for  holding 
liquids  must  have  strength  to  withstand  their  gravity  pres- 
sure, and  that  only.  Vessels  for  holding  gases  must  have 
strength  to  withstand  the  force  developed  by  compressing 
them,  and  practically  that  only,  for  the  gravity  pressure 
is  negligibly  small. 

For  example,  suppose  a  cylindrical  steel  tank  for  holding  oxygen 
or  hydrogen  to  be  12  in.  in  diameter  and  4  ft.  high,  and  to  contain 
either  of  these  gases  under  a  pressure  of  200  Ib.  per  sq.  in.  The 
thrust  against  the  top  of  the  tank  would  be  over  22,000  Ib.,  while  the 
thrust  on  the  bottom  would  exceed  this  only  by  the  weight  of  the  gas, 
which,  if  it  was  oxygen,  would  be  about  3.4  Ib.,  or,  if  hydrogen,  less 
than  I  Ib. 

The  gravity  pressure  of  gases  is,  in  general,  negligibly 
small;  with  the  single  exception  of  the  atmosphere;  and 
that  only  on  account  of  its  great  height. 

61.  The  Elastic  Force  of  Fluids.  —  All  fluids  offer  resist- 
ance to  compression,  —  liquids  a  very  great  resistance  to 
any  appreciable  compression,  gases  comparatively  little. 
The  resisting  pressure  developed  within  a  fluid  by  compres- 
sion is  called  its  elastic  force.  This  is  always  equal  to  the 


LAWS  OF  GASES  53 

applied  pressure;  in  fact,  it  is  the  applied  pressure,  trans- 
mitted throughout  the  fluid  in  accordance  with  Pascal's 
law.  All  fluids  expand  to  their  original  volume  after  com- 
pression, when  the  added  pressure  is  removed.  In  other 
words,  fluids  have  perfect  elasticity  of  volume. 

To  illustrate:  The  pressure  at  a  depth  of  3  miles  in  the  ocean  is, 
in  round  numbers,  7000  Ib.  per  sq.  in.  The  loss  of  volume  under  this 
pressure  is  approximately  2%.  If  a  bottle,  filled  and  sealed  at 
this  depth,  were  brought  to  the  surface,  it  would  burst,  unless  it 
was  capable  of  withstanding  the  elastic  force  of  7000  Ib.  per  sq.  in., 
which  the  water  would  exert  in  consequence  of  being  compressed. 
Similarly,  each  cubic  centimeter  of  the  atmosphere  near  sea-level  is 
subjected  to  a  pressure  of  more  than  1000  g.  against  each  of  its  sides 
(a  force  approximately  800,000  times  as  great  as 
its  weight),  and  it  reacts  with  an  equal  pressure 
(elastic  force)  in  resisting  further  compression. 

Surprisingly  great  as  the  elastic  force  of  gases 
is  in  comparison  with  their  weight,  the  contrast 
is  even  greater  with  liquids.  While  water  is  about 
800  times  as  dense  as  air  under  ordinary  condi- 
tions, the  resistance  that  it  offers  to  a  given  com- 
pression is  about  20,000  times  as  great  as  thkt  of 
air.  It  must  be  remembered,  however,  that  the 
elastic  force  in  a  body  of  liquid  ceases  when  the 
liquid  has  expanded  to  a  definite  volume,  while  in 
a  mass  of  gas  it  continues  (with  diminishing  in- 
tensity) however  great  the  expansion  may  be. 

52.    Measurement  of  Gas  Pressure. — An  in-      FlG-  32.  — Open 
strument  for  measuring  the  elastic  force  or  pres- 
sure of  a  gas  in  a  closed  space  is  called  a  pressure  gage  or  ma- 
nometer.    Manometers  are  made  in  a  variety  of  forms,  adapted 
to  the  amount  of  pressure  which  they  are  intended  to  measure. 

The  open-tube  manometer  (Fig.  32)  is  commonly  used  for  measur- 
ing pressures  only  slightly  greater  or  less  than  that  of  the  atmosphere. 
It  consists  of  a  glass  U-tube  partly  filled  with  water  or  mercury,  with 
a  rubber  tube  attached  to  one  arm  for  making  connection  with  the 
vessel  in  which  the  gas  is  contained,  and  a  scale  for  measuring  the 


54  STATICS  OF   GASES 

height  of  the  liquid  in  the  two  arms.  When  such  a  manometer  is 
connected  with  the  gas  pipes  of  a  building  and  the  gas  turned  on,  the 
liquid  is  pushed  down  in  the  arm  in  which  the  gas  is  admitted.  The 
pressure  of  the  gas  upon  the  surface  a  is  equal  to  the  pressure  at 
the  same  level,  c,  in  the  other  arm;  and  we  know  that  the  pressure  at 
c  is  the  sum  of  the  atmospheric  pressure  on  b  and  the  gravity  pressure 
of  the  column  of  liquid,  be.  Hence  the  pressure  of  the  gas  exceeds 
the  pressure  of  the  air  by  an  amount  equal  to  the  gravity  pressure 
of  the  liquid  column  be.  This  excess  of  pressure  can  be  computed 
in  grams  per  square  centimeter,  if  so  desired,  from 
the  measured  height  of  the  column  and  the  known 
density  of  the  liquid.  To  find  the  whole  pressure  of 
the  gas,  the  pressure  of  the  atmosphere  (determined 
by  the  barometer)  must  be  added. 

The  vacuum  gage  (Fig.  33)  is  used  for  measuring 
gas  pressure  in  partially  exhausted  vessels.  It  is  a 
closed-tube  manometer,  having  no  air  or  other  gas 
in  the  closed  arm.  In  construction  and  action  it  is 
like  the  siphon  barometer,  with  the  exception  that 
its  closed  arm  is  commonly  much  shorter,  and  the 
mercury  completely  fills  this  arm  when  under  atmos- 
pheric pressure.  While  the  air  or  other  gas  is  be- 
ing pumped  from  a  vessel  to  which  a  vacuum  gage  is  attached, 
the  mercury  continues  to  fill  the  closed  arm  for  some  time,  if  the  origi- 
nal pressure  was  more  than  sufficient  to  sustain  the  full  height  of  the 
column.  It  is  only  after  Jhe  mercury  begins  to  fall  that  the  difference 
of  level  measures  the  pressure.  When  the  mercury  stands  at  the 
same  level  in  the  two  arms,  the  inclosed  space  with  which  the  gage 
is  connected  is  a  perfect  vacuum. 

The  steam  gage  (Fig.  34)  is  used  for  measuring  high  pressures, 
as  in  steam  boilers,  and  in  tanks  of  compressed  air  and  other  gases. 
A  is  a  cock  in  a  small  pipe  connecting  the  gage  with  the  tank  or 
boiler;  B  is  a  bent  tube  of  elliptical  section,  as  shown  at  the  bottom 
of  the  figure,  one  end  of  which  is  joined  to  the  cock  and  the  other 
closed  and  free  to  move.  The  free  end  works  a  sector  D  and  pinion 
E,  by  means  of  the  connecting  link  C.  D  is  pivoted  at  its  center, 
and  E  carries  a  pointer,  F,  which  revolves  with  it.  When  the  pres- 
sure within  the  bent  tube  increases,  the  cross-section  of  the  tube 


LAWS   OF   GASES 


55 


Section  of  tube  B. 
FIG.  34. — Dial  Pressure  Gage. 


becomes  more  nearly  circular,  and  the  tube  tends  to  straighten. 
This  causes  the  sector  to  move,  and  the  pointer  to  indicate  the 
pressure  on  the  scale. 

53.  Units  of   Fluid   Pres- 
sure.— The  pressure  of  a  gas 
is  often  expressed  in  terms  of 
the  height  of  -the  mercury  or 
water  column  that  it  can  sup- 
port in  a  manometer,  e.g.  we 
speak  of  a  pressure  of  so  many 
centimeters  or  millimeters  of 
mercury  in  the  receiver  of  an 
air  pump .   Great  pressures  are 
generally  expressed  in  atmos- 
pheres, as  a  pressure  of  500 
atmospheres.  An  atmosphere 
is  equal  to  the  pressure  of  a 

column  of  mercury  76  cm.  high.  This  unit  is  derived 
from  the  average  pressure  of  the  atmosphere  at  sea-level. 
It  is  constant,  and  wholly  independent  of  the  existing 
barometric  pressure  at  any  time  or  place. 

The  principal  units  of  fluid  pressure  are  the  gram  per 
square  centimeter,  the  pound  per  square _inch,  the  centimeter 
or  the  inch  of  mercury,  the  centimeter  or  the  inch  of  water, 
and  the  atmosphere.  The  student  should  be  able  to  com- 
pute a  pressure  in  terms  of  any  one  of  these  units,  when  its 
value  in  terms  of  any  other  one  of  them  is  known. 

54.  Boyle's  Law.  —  For  twenty  years  or  more  after  the 
barometer  was  invented  (1643),  the  experimental  study  of 
atmospheric  pressure  and  the  general  behavior  of  the  air 
under  different  pressures  was  diligently  pursued  by  physi- 
cists in  the  principal  countries  of  Europe.     The  air  pump, 


56  STATICS    OF    GASES 

invented  by  Guericke  in  1650,  opened  up  a  new  field  of  in- 
vestigation that  excited  wonder  and  curiosity.    In  England 

the  mechanics  of  the  air  was 
first  studied  by  Robert  Boyle 
(1627-1691).  He  was  especially 
interested  in  measuring  what 
he~~termed  "the  spring  of  the 
air"  under  different  degrees  of 
compression,  or  the  relation  be- 
tween the  different  volumes  of 
the  same  mass  of  air  and  the 
corresponding  pressures  exerted 
by  (and  upon)  it. 

For  this  purpose  he  prepared  a 
|~aa  bent  glass  tube  (Fig.  35),  having 
the  shorter  arm  closed  and  the 
longer  arm  open.  Having  poured 
in  a  little  mercury,  he  adjusted  the 
quantity  of  air  in  the  closed  arm 
so  that  the  mercury  stood  at  the 
same  level  in  both  arms  (A,  Fig. 
35).  The  air  in  the  closed  tube  was  then  under  the  same  pressure 
as  the  outside  air,  which  the  barometer  showed  to  be  equal  to  29 
inches  of  mercury.  Having  measured  the  length  of  the  air  column 
in  the  closed  tube,  he  poured  in  more  mercury  till  the  length  of  the 
air  column  was  reduced  exactly  one  half.  He  then  observed  "not 
without  delight  and  satisfaction"  that  the  mercury  stood  29  inches 
higher  in  the  open  arm  than  in  the  other  (B,  Fig.  35).  Since  the 
added  pressure  of  this  column  of  mercury  was  equal  to  the  pressure 
of  the  atmosphere,  the  pressure  upon  the  confined  air  had  been 
doubled  in  reducing  its  volume  one  half.  Other  measurements,  taken 
with  different  pressures,  showed  the  same  relation,  i.e.  the  volume 
of  the  confined  air  decreases  at  the  same  rate  that  the  pressure  upon 
it  increases,  and  vice  versa. 

Later  and  more  accurate  investigations  have  shown  that 
this  relation  holds  very  approximately  (not  exactly)  for 


FIG.  35.  —  Boyle's  Experiment. 


LAWS  OF  GASES  57 

all  gase.s,  until  the  pressure  is  so  great  that  the  gas  is  not 
far  from  liquefying.  This  behavior  is  summed  up  in 
Boyle's  law:  The  volume  of  a  given  mass  of  any  gas  varies 
inversely  as  the  pressure  upon  it,  provided  the  temperature 
remains  constant. 

If  Vi  denotes  the  volume  of  a  mass  of  gas  when  the 
pressure  upon  it  is  P1}  and  V2  its  volume  under  a  differ- 
ent pressure  P2,  the  temperature  remaining  the  same,  the 
algebraic  statement  of  Boyle's  law  will  be 

Pi:P,::7,:7i. 

From  this  proportion  it  follows  that  PiVi  =  PZV2'}  i.e. 
Sit  a  constant  temperature,  the  product  of  the  pressure  and 
volume  of  a  given  body  of  gas  is  constant.  The  volume 
of  a  gas  does  not  vary  inversely  as  the  pressure  if  its 
temperature  changes;  for  a  change  of  temperature  will 
itself  produce  either  a  change  of  pressure  or  a  change  of 
volume. 

55.  Effect  of  Pressure  on  the  Density  of  a  Gas.  —  When 
a  gas  is  compressed,  its  density  increases  in  proportion  to 
the  decrease  of  its  volume;  but  the  pressure  also  increases 
in  proportion   to  the  decrease  of  volume,  provided  the 
temperature  remains  constant  (Boyle's  law).     Hence  the 
density  of  a  gas   at  a  constant  temperature  is  directly 
proportional  to  the  pressure  upon  it. 

56.  Height  and  Density  of  the  Atmosphere.  —  The  density  of 
the  air  is  less  at  higher  altitudes  because  the  pressure  is  less.     The 
height  to  which  the  last  scanty  remnant  of  the  atmosphere  extends 
is  unknown,  but  it  is  variously  estimated  at  from  100  to  200  miles. 
It  is  known  to  extend  above  fifty  miles;  yet  the  density  decreases 
so  rapidly  that  the  pressure  at  a  height  of  3.4  miles  is  only  one  half 
as  great  as  at  sea-level  (Fig.  36) ;  from  which  it  follows  that  one  half 
of   the   air  lies   below   the   latter   elevation.     (Why?)     Men   have 
ascended  to  higher  altitudes  than  this  upon  mountains,  and,  in  a  few 


58  STATICS  OF  GASES 

instances,  to  a  height  of  6  or  7  miles  in  balloons.  At  high  altitudes 
the  mass  of  air  taken  into  the  lungs  with  each  breath  is  greatly  re- 
duced, and  breathing  must  be  more  rapid  to  make  up  the  deficiency. 
A  very  little  exertion  brings  on  a  violent  struggle  for  air,  accompanied 


FIG.  36.  —  Diagram  of  the  Atmosphere.  The  region  of  convection  currents, 
clouds,  and  storms  extends  to  a  height  of  about  10  km.  (6.21  mi.).  Above  this 
is  the  isothermal  layer,  extending  to  an  unknown  height.  It  is  exceedingly  dry 
and  cold.  Above  the  "  limit  of  twilight "  the  air  is  too  rarefied  to  reflect  sun- 
light to  the  earth;  but  clouds  of  fine  dust  from  Krakatoa  floated  in  this  region 
for  two  years  after  the  eruption  in  1883.  These  were  seen  at  night  by  re- 
flected sunlight;  hence  the  term  "noctilucent."  Higher  still,  the  existence  of 
a  scanty  atmosphere  is  demonstrated  by  the  light  of  falling  meteors  (due  to 
the  friction  of  the  air),  and,  last  of  all,  by  the  aurora. 

by  a  feeling  of  suffocation.  The  insufficient  pressure  upon  the  body 
and  the  intense  cold  are  further  disagreeable  and  even  dangerous 
features  of  very  high  balloon  ascensions. 

If  the  atmosphere  were  of  the  same  density  throughout  as  at  sea- 
level,  it  would  extend  only  to  a  height  of  about  five  miles, 


LAWS  OF  GASES  59 

57.  Buoyancy  of  the  Air.  —  The  law  of  buoyancy  holds 
for  bodies  in  air  as  well  as  for  bodies  immersed  in  liquids, 
and  for  the  same  reason  (Art.  37).     A  body  in  air  is  buoyed 
up  by  a  force  equal  to  the  weight  of  the  air  displaced  by  it. 
A  cork  rises  to  the.  surface  in  water  and  a  balloon  rises  in 
air  under  like  conditions.     In  both  cases  the  weight  of  the 
body  is  less  than  the  buoyant  force.     The  gas  in  a  balloon 
does  not  of  itself  exert  a  lifting  force.     On  the  contrary,  it 
is  subject  to  the  force  of  gravity,  as  all  matter  is;  but  its 
weight  is  much  less  than  that  of  the  displaced  air,  leaving 
an  excess  of  buoyancy  more  than  sufficient  to  support  the 
weight  of  the  balloon  and  the  occupants  of  the  car,  just  as  a 
cork  may  carry  up  through  water  a  pebble  that  is  tied  to  it. 

The  buoyant  force  of  the  air  upon  solids  and  liquids  is  always 
very  small  in  comparison  with  their  weight.  Upon  water  it  is  about 
slo  of  the  weight,  and  upon  lead  less  than  Woo;  upon  the  body  of  a 
grown  person  it  is  about  3  ounces.  The  ordinary  circumstances  of 
life  afford  no  means  of  detecting  this  buoyant  force  upon  solids 
and  liquids.  (A  bird,  a  kite,  or  a  feather  is  not  sustained  in  air  by 
buoyancy,  but  by  pressure  due  to  the  inertia  of  jail.)  Upon  gases, 
however,  the  buoyancy  is  relatively  large.  It  exceeds  the  true 
weight  of  any  gas  less  dense  than  air,  e.g.  hydrogen  and  illuminating 
gas,  and  all  such  gases  tend  to  rise.  The  weight  of  a  gas  is  always 
expressed  as  its  true  weight,  i.e.  its  weight  in  a  vacuum,  the  buoyancy 
of  the  air  being  much  too  large  to  be  disregarded. 

58.  Summary  of  the  Laws  of  Gases.  —  The  laws  of  gases 
include  Pascal's  law  (Art.  34),  Boyle's  law  (Art.  54),  and  the 
law  of  buoyancy  or  Archimedes'  principle  (Art.  37).     To 
these  must  be  added  the  law  of  Charles  (Art.  206),  which  ex- 
presses the  effect  of  temperature  upon  the  volume  of  a  gas. 

The  first  two  laws  of  Art.  29  hold  for  gases;  but  they  are  included 
in  Pascal's  lam-  Gravity  pressure  in  gases  is  so  small  as  to  be  practi- 
cally negligible,  eja^pt_in__the_case  of  the  atmosphere,  and  in  this 
case  it  is  not  proportional  to  the  depth,  since  the  density  of  the 
atmosphere  is  not  uniform, 


60  STATICS  OF  GASES 

PROBLEMS 

1.  In  ascending  a  mountain  will  the  fall  of  the  barometer  during  each 
thousand  feet  of  the  ascent  be  greater  or  less  than  for  the  preceding  thousand 
feet?    Why? 

2.  Describe  the  process  by  which  the  air  enters  the  lungs  in  breathing. 
Criticize  the  expression  "drawing  in  a  breath." 

3.  (a)  At  what  depth  in  fresh  water  is  the  gravity  pressure  equal  to  one 
atmosphere?     (6)  At  what  depth  in  salt  water? 

4.  From  what  depth  in  freshwater  must  a  bubble  of  gas  start  in  order 
that  its  volume  may  be  doubled  by  the  time  it  reaches  the  surface? 

6.   A  bag  of  feathers  and  a  piece  of  iron  weigh  exactly  a  pound  each  in 
air.     Which  has  the  greater  true  weight?     Which  has  the  greater  mass? 

6.  Is  the  buoyant  force  upon  the  body  of  a  swimmer  affected  by  the 
greater  or  less  inflation  of  his  lungs? 

7.  A  cubic  decimeter  of  gas  is  under  a  pressure  of  100  cm.  of  mercury. 
What  will  be  its  volume  at  the  same  temperature  under  a  pressure  of  30  cm. 
of  mercury? 

8.  A  liter  of  gas  is  taken  under  a  pressure  of  one  atmosphere.    What 
will  be  its  volume  at  the  same  temperature  under  a  pressure  of  100  cm.  of 
mercury? 

9.  Two  liters  of  gas  under  a  pressure  of  one  atmosphere  will  have  what 
volume  when  the  pressure  is  reduced  to  900  g.  per  sq.  cm.? 

10.  Compute  the  height  to  which  the  earth's  atmosphere  would  extend  if  it 
had  the  same  density  at  all  altitudes  as  at  sea-level,  assuming  that  density  to 
be  .0012  g.  per  can.,  and  taking  the  pressure  at  sea-level  as  76  cm.  of  mercury. 

11.   The  length  of  the  air  column  in 
the  closed  arm  of  a  Boyle's  law  appa- 
ratus is  25  cm.  when  the  mercury  stands 
20  cm.  higher  in  the  closed  arm  than 
in  the  open  arm.    What  will  be  the  length  of  the  air  column 
when  the  mercury  stands  30  cm.  higher  in  the  open  arm  than 
in  the  closed  arm,  the  atmospheric  pressure  being  75  cm.? 

12.   An  open  manometer  (Fig.  37)  is  connected  with  a 
vessel  containing  air  at  o°  C.     The  mercury  stands  15  cm. 
FIG.  37.      higher  in  the  outer  arm  of  the  gage  than  in  the  inner  arm, 
and  the  barometer  reads  75  cm.     Compute  the  density  of  the  air  in  the 
vessel.    (The  density  of  air  at  o°  C.  is  .001293  g-  per  ccm.  under  a  pressure 
of  76  cm.  of  mercury.) 


THE  MECHANICS  OF  FLUIDS  61 

III.  APPLICATIONS  OF  THE  MECHANICS  OF  FLUIDS 

59.  The  Air  Pump.  —  The  air  pump,  designed  for  remov- 
ing air  or  other  gas-  from  a  closed  vessel,  was  invented  by 
Otto  von  Guericke.  It  has  been  improved  at  various  times, 

and  is  now  made  in  many 
forms  differing  greatly  from 
one  another  in  details  of  con- 
struction and  in  effectiveness. 


FIG.  38.  —  Air  Pump  with  Pressure  Gage. 

Figure  38  represents  one  of  the  older  forms,  involving  only  the 
earliest  and  simplest  principles.  The  pump  consists  of  a  metal 
cyb'nder  in  which  fits  an  air-tight  piston  operated  by  the  handle. 
There  are  two  valves,  namely,  the  piston,  vate  a  and  the  inlet  valve 
b,  the  latter  covering  the  end  of  the  tube  that  leads  to  the  bottom  of 
the  cylinder.  The  valves  open  upward  only,  as  shown  in  the  figure. 
The  simplest  form  of  valve  consists  of  a  piece  of  thin  leather  or  oiled 
silk,  placed  so  as  to  cover  the  hole  and  fastened,  at  one  edge.  The 
valve  closes  the  opening  air  tight  whejLpressed  against  itf  and  leaves 
it  open  when  pushed  in  the  opposite  direction.  The  pump  is  con- 
nected by  the  tube  to  an  opening,  0,  at  the  center  of  a  flat  metal  plate, 
PQ,  upon  which  stands  a  receiver,  R. 

The  action  of  the  pump  is  as  follows.  Suppose  the  piston  to  be 
at  rest  at  the  bottom  of  the  cylinder.  Both  valves  will  be  closed, 
being  held  down  by  their  weight.  During  the  up-stroke  of  the  piston, 


62 


STATICS  OF   GASES 


the  small  amount  of  air  beneath  it  expands  and  fills  the  increased 
space,  and  its  pressure  decreases  proportionally.  The  atmospheric 
pressure  upon  the  top  of  valve  a  being  now  greater  than  the  pressure 
from  beneath,  this  valve  is  firmly  closed.  When  the  downward 
pressure  upon  b  is  sufficiently  diminished,  the  pressure  of  the  air  in 
the  tube  beneath  this  valve  lifts  it,  permitting  some  of  the  air  in  the 
receiver  to  escape  into  the  space  below  the  piston.  As  soon  as  the 
piston  stops  rising,  the  lower  valve  is  closed  by  its  own  weight.  On 
pushing  the  piston  down,  the  air  beneath  it  is  compressed.  This 
air  can  not  escape  through  the  lower  valve,  since  the  increased  pres- 
sure only  closes  this  valve  more  tightly.  When  the  amount  of  com- 
pression is  such  that  the  density  of  the  confined  air  is  slightly  greater 
than  that  of  the  atmosphere,  the  upper  valve  is  forced  open,  permit- 
ting the  air  to  escape. 

These  processes  are  repeated  with  every  stroke  of  the  piston, 
thus  gradually  removing  the  air  from  the  receiver.  The  limit  of 
possible  exhaustion  is  reached  when  the  pressure  of  the  air  remaining 
in  the  receiver  is  insufficient  to  lift  the  lower  valve,  or  when  the 

quantity  of  air  that  enters  the  cyl- 
inder with  the  up-stroke  is  so  small 
that  it  can  not  be  compressed 
enough  to  raise  the  upper  valve. 
The  newer  forms  of  air  pumps 
do  not  depend  upon  the  pressure 
of  the  air  for  operating  the  valves, 
and  are  therefore  capable  of  pro- 
ducing a  more  nearly  perfect 
vacuum.  They  are  commonly 
provided  with  metal  valves, 
which  are  operated  automatically 

by  a  simple  mechanism  attached 
FIG.  39.  —  Exhaust  and  Compression        J 


Pump. 


to  the  piston  or  to  the  .piston-rod. 


60.  Compression  Pumps.  —  If  the  valves  of  the  pump 
shown  in  Fig.  38  were  made  to  open  downward,  the  ac- 
tion of  the  pump  would  be  reversed,  and  it  would  force 
air  into  the  receiver.  It  would  thus  become  a  compres- 
sion pump. 


THE  MECHANICS  OF  FLUIDS 


FIG.  40.  —  Hand 
Bellows. 


A  pump  may  be  constructed  so  as  to  serve  both  purposes,  as  shown 
in  Fig.  39.  The  air  (or  other  gas)  enters  the  pump  through  the  valve 
A,  which  opens  inward,  and  is  driven  out 
through  C,  which  opens  outward.  Hence  if 
a  closed  vessel  is  attached  at  C,  air  will  be 
forced  into  it;  if  attached  at  A,  the  air  will 
be  exhausted  from  it.  {Describe  the  action 
in  detail.) 

A  bicycle  pump  is  a  simple  form  of  compression  pump.  (Ex- 
amine one  and  describe  its  action.)  The  bellows  is  a  simple  device 
for  supplying  a  large  volume  of  air  under  moderate  pressure.  If  it 
has  only  one  compartment,  as 
in  the  hand  bellows  (Fig.  40), 
the  flow  is  intermittent;  if  it 
has  two  compartments,  as  in 
the  blacksmith's  bellows  and  the 
organ  bellows,  the  flow  is 
continuous. 

Powerful  compression  pumps, 
operated  by  engines,  are  much 
used  for  compressing  air  on  a 
large  scale  for  various  industrial 
purposes,  as  for  operating  air 
brakes,  air  drills,  pneumatic  ham- 
mers, etc.,  and  for  keeping  diving 
bells,  caissons,  and  tunnels  sup- 
plied with  air  while  work  is 
being  done  under  water.  (Fig.  41 .) 

|f 

61.  The  Lifting  Pump.— The 
common  lifting  or  suction  pump 
(Fig.  42)  is  one  of  the  oldest 
mechanical  devices,  its  use  dat- 
ing from  the  fourth  century  B.C.  It  is  similar  to  the  air  pump  in  its 
construction  and  action.  The  valves  open  upward,  as  shown  in  the 
figure.  A  pipe  extends  from  the  cylinder  or  barrel  of  the  pump  to 
some  distance  below  the  surface  of  the  water  in  the  well  or  cistern. 
The  piston  is  operated  by  means  of  a  handle,  acting  as  a  lever.  The 
pump  at  first  acts  as  an  air  pump  in  exhausting  the  air  from  the 
pipe.  While  this  is  taking  place,  the  pressure  of  the  air  remaining 


FIG.  41. — Diving  Bell. 


64 


STATICS  OF  GASES 


in  the  pipe  decreases,  and  the  greater  pressure  of  the  atmosphere 
upon  the  water  in  the  well  pushes  water  up  into  the  pipe,  just  as 
mercury  is  forced  up  and  sustained  in  a  barometer  tube.  After 
the  pump  is  filled  with  water  in  this  manner,  the  closing  of  the 
lower  valve  during  the  down-stroke  of  the  piston  prevents  the  return 
of  the  water  into  the  pipe.  At  the  same  time  the  valve  in  the  piston  is 

forced  open,  and  tthe 
water  flows  through  it 
into  the  space  above. 
At  the  beginning  of 
the  up-stroke  the 
valve  in  the  piston 
falls,  and  the  water 
above  it  is  lifted  out. 
Since  the  entire 
pressure  of  the  at- 
mosphere at  sea-level 
can  sustain  a  column 
of  water  only  to  a 
height  of  about  10.3  meters  (34  ft.),  the  lower  valve  would  have  to 
be  within  that  distance  of  the  water  in  the  well  even  if  the  pump 
were  capable  of  producing  a  perfect  vacuum.  The  actual  limit  of 
distance  for  a  good  pump  is  about  26  feet. 


FIG.  42.  —  Lift  or  Suction  Pump. 


62.  The  Force  Pump.  —  In  the  force  pump  the  second  valve  is 
placed  at  the  entrance  to  the  discharge  pipe,  B  (Fig.  43).  There  is 
no  valve  in  the  piston.  The  action 
of  the  pump  during  the  up-stroke 
of  the  piston  is  the  same  as  in  the 
lifting  pump.  (Which  valve  is  open? 
Which  closed?)  With  the  down- 
stroke  of  the  piston  the  water  in 
the  barrel  of  the  pump  is  forced 
into  the  discharge  pipe.  The  height 
to  which  water  can  be  forced  in 
this  pipe  depends  only  upon  the 
strength  of  the  pump,  being  in  no 
way  affected  by  atmospheric  pressure.  FIG.  43.  —  Force  Pump. 


THE  MECHANICS  OF  FLUIDS 


A  force  pump  is  generally  provided  with  an  air  chamber, 
which  is  connected  with  -the  discharge  pipe.  During  the  down- 
stroke  of  the  piston  some  of  the  water  is  forced  into  this  chamber, 
and  compresses  the  air.  During  the  up-stroke  the  expanding  air 
drives  the  water  out  of  the  chamber,  thus  maintaining  a  contin- 
uous flow. 


FIG.  44.  —  Sectional  View  of  a  Steam  Pump. 

Force  pumps  are  used  to  raise  water  to  a  higher  level  in  filling 
tanks,  reservoirs,  and  stand-pipes,  and  to  deliver  it  under  great 
pressure,  as  in  hydraulic  presses  (Art.  35)  and  fire  engines.  They 
are  operated  by  hand,  by  windmills,  and  by  engines  (Fig.  44). 

63.  The  Siphon.  —  A  bent  tube  or  pipe  for  conveying  liquids 
over  an  elevation  from  a  higher  to  a  lower  level  is  called  a  siphon 
(Fig.  45,  a  and  b).  Either 
a  rigid  or  a  flexible  tube 
will  serve  the  purpose.  To 
start  a  small  siphon  it  may 
be  held  with  the  bend  down 
and  filled,  then,  with  a  fin- 
ger over  each  end,  inverted 


FIG.  450.— The 
Siphon. 


FIG.  456.  — 
Aspirating 
Siphon. 


and  placed  in  position;  or  it  may  first  be  placed 
in    position,    and    the     air    then    exhausted    by 
"applying  the  mouth  to  the  lower  end.      The  liquid  will  continue  to 
flow  as  long  as  one  end  of  the  siphon  is  covered  by  it  and  the  other 


66 


STATICS  OF  GASES 


end  is  below  the  level  of  its  surface  (i.e.  below  ab  in  the  figure); 
but  if  the  outlet  of  the  siphon  is  also  immersed,  the  flow  will  cease 
as  soon  as  the  liquid  reaches  the  same  level  in  the  two  vessels. 

To  explain  the  action  of  the  siphon  we  may  suppose  it  to  be  stopped 
by  closing  the  outlet,  c,  with  the  finger.  The  liquid  will  then  be  at 
rest,  and  the  laws  of  pressure  for  liquids  in  equilibrium  will  hold. 
At  points  a  and  b  in  the  tube,  on  a  level  with  the  surface  of  the 
liquid,  the  pressure  is  the  same  as  that  of  the  atmosphere.  (How 
do  we  know?)  The  pressure  at  c  is  equal  to  this  plus  the  gravity 
pressure  of  the  liquid  column  be.  When  the  finger  is  removed,  the 
only  upward  pressure  at  c  is  that  of  the  atmosphere,  which  leaves 
the  gravity  pressure  of  the  column  be  unbal- 
anced. This  unbalanced  downward  pres- 
sure, acting  on  the  liquid  in  the  siphon, 
causes-  it  to  flow.  The  liquid  would  part 
at  the  top  and  run  out  at  both  ends,  leav- 
ing the  siphon  empty,  if  it  were  not  for  the 
pressure  of  the  atmosphere,  which,  acting 
inward  at  both  ends,  holds  the  liquid  in  a 
continuous  column  and  compels  it  all  to 
flow  in  the  same  direction. 

The  siphon  is  useful  in  drawing  liquids 
from  vessels  where  pouring  is  inconvenient, 
and  in  removing  the  upper  part  of  a  liquid 
when  the  lower  part  is  of  a  different  kind 
or  quality  or  contains  sediment.  It  is  usu- 
ally provided  with  a  suction  tube  (Fig.  456) 
for  starting  the  flow  without  permitting  any 
of  the  liquid  to  enter  the  mouth.  To  start  such  a  siphon  the  lower 
end  is  closed  while  the  air  is  exhausted  through  the  suction  tube. 
64.  The  Balloon.  —  Since  the  buoyant  force  upon  a  balloon  is 
equal  to  the  weight  of  the  displaced  air,  its  amount  depends  only 
upon  the  size  of  the  balloon.  The  carrying  capacity  of  a  balloon  is 
determined  by  the  difference  between  the  buoyant  force  and  its  own 
weight,  including  the  true  weight  of  the  gas  with  which  it  is  filled. 
Hence  it  is  an  advantage  to  use  hydrogen,  which  is  the  least  dense 
of  gases;  but  illuminating  gas  is  generally  used,  as  it  is  cheaper  and 
more  easily  obtained.  The  first  balloons  were  inflated  with  hot  air.  • 
If  a  balloon  is  not  fully  inflated  at  the  start,  the  gas  within  it 


Baling 
Bags 


Guide  or 
Trail  Rope 


FIG.  46. —Balloon. 


THE  MECHANICS  OF  FLUIDS  67 

expands  as  the  balloon  rises,  in  consequence  of  the  diminishing 
atmospheric  pressure  upon  it.  As  long  as  there  is  room  for  this 
expansion,  the  buoyant  force  remains  constant,  for  the  increase  in 
the  volume  of  the  displaced  air  offsets  the  decrease  in  its  density. 
As  a  balloon  rises  after  becoming  fully  distended,  the  buoyant  force 
decreases  until  it  is  no  greater  than  the  true  weight  of  the  balloon 
and  all  it  carries.  It  then  ceases  to  rise,  unless  lightened  by  throw- 
ing out  sand,  a  supply  of  which  is  carried  for  this  purpose.  When 
the  aeronaut  wishes  to  descend,  he  opens  a  valve  at  the  top  of 
the  balloon,  and  some  of  the  gas  escapes. 

PROBLEMS 

1.  Over  how  great  an  elevation  can  water  be  siphoned?     Why?     Over 
how  great  an  elevation  can  mercury  be  siphoned?     Would  a  siphon  work  in 
a  vacuum?     Explain.  •     .     . 

2.  (a)  At  ordinary  temperatures  and  under  a  pressure  of  one  atmosphere 
a  cubic  meter  of  air  weighs  about  1.2  kg.,  a  cubic  meter  of  hydrogen  about 
.083  kg.,  and  a  cubic  meter  of  illuminating  gas  about  .74  kg.     Assuming 
these  values,  what  is  the  buoyant  force  upon  a  balloon  containing  500  cu.  m. 
of  hydrogen?,     (b)  How  great  a  weight  will  this  buoyant  force  sustain  in 
addition  to  the  weight  of  the  hydrogen? 

3.  With  what  volume  of  illuminating  gas  must  a  balloon  be  filled  to  rise, 
if  the  empty  balloon,  the  car,  and  the  occupants  together  weigh  500  kg.? 

4.  Will  the  true  weight  of  a  body  be  greater  or  less  than  its  weight  in 
air  when  weighed  on  an  equal-arm  balance  with  brass  weights  (a)  if  the 
density  of  the  body  is  the  same  as  that  of  brass?  (b)  if  its  density  is  less? 
(c)  if  its  density  is  greater? 

5.  The  human  heart  is  a  pair  of  force  pumps.     Consult  any  physiology 
for  a  description  of  it  and  its  action.     Compare  its  construction  and  action 
with  that  of  a  force  pump  as  described  above.     How  is  a  continuous  flow 
of  blood  maintained  in  the  arteries;  in  other  words,  what  corresponds  to  the 
air  chamber  of  an  ordinary  force  pump? 


CHAPTER  V 

STATICS  OF  SOLIDS 

65.  Introduction.  —  Every  portion  of  matter  on  the 
earth,  whether  at  rest  or  in  motion,  is  constantly  acted 
upon  by  a  number  of  forces.  A  body  is  always  subject 
to  the  action  of  gravity  (its  weight) ;  and,  except  in  the  very 
unusual  case  of  a  body  falling  freely  in  a  vacuum,  it  is  at 
every  instant  acted  upon  by  one  or  more  other  forces  as 
well.  When  a  body  at  rest  remains  at  rest,  or  a  body  in 
motion  continues  with  unchanging  motion,  we  conclude 
without  further  evidence  that  the  joint  effect  of  all  the 
forces  then  acting  upon  it  is  nil  or  zero,  so  far  as  the  body 
as  a  whole  is  concerned;  and  the  forces  thus  acting  are 
said  to  be  in  equilibrium,  or  to  constitute  a  set  of  balanced 
forces  (Art.  10). 

Many  examples  have  been  studied  in  the  previous  chapters.  The 
weight  of  a  floating  body  and  the  buoyant  force  exerted  upon  it  by 
the  liquid  in  which  it  is  floating  constitute  a  pair  of  balanced  forces. 
In  this  case,  as  in  many  others,  the  pressure  of  the  air  may  be  left 
out  of  account,  since  its  net  result  is  a  buoyant  force  which  is  negli- 
gibly small.  Forces  not  in  equilibrium  have  also  been  incidentally 
considered,  e.g.  the  forces  acting  on  a  cork  released  under  water. 
The  buoyant  force  in  this  case. is-  greater  than  the  weight  of  the  cork; 
and  the  excess  of  buoyancy,  being  unbalanced,  pushes  the  cork  to 
the  surface. 

It  is  only  under  certain  special  conditions  that  a  set  of 
forces,  acting  together  on  a  body,  will  be  in  equilibrium, 
and  it  is  with  these  conditions  or  relations  that  we  are  prin- 
cipally concerned  in  the  present  chapter.  This  study  yields 

68 


CONCURRENT  FORCES  69 

many  interesting  and  important  facts  concerning  the  ordi- 
nary behavior  of  bodies,  and  affords  at  least  some  insight 
into  the  mechanical  principles  involved  in  the  arch,  the 
truss,  the  suspension  cable,  and  other  structural  forms 
employed  in  building  houses,  bridges,  etc. 

In  passing  from  the  mechanics  of  fluids  to  the  mechanics 
of  solids,  it  should  be  noted  that  forces  acting  on  solids 
may  be  and  often  are  concentrated  practically  at  points; 
hence,  as  a  rule,  we  shall  not  have  occasion  to  consider 
areas  or  force  per  unit  area. 

I.   CONCURRENT  FORCES 

66.  Equilibrium  of  Two  Forces.  —  The  relations  that 
must  exist  among  two  or  more  forces  in  order  that  they 
may  balance  each  other  are  referred  to  as  the  conditions 
necessary  for  equilibrium,  or,  simply,  the  conditions  of 
equilibrium. 

The  conditions  necessary  for  the  equilibrium  of  two  forces  were 
briefly  considered  for  solids  in  Art.  10,  and  they  have  been  exempli- 
fied repeatedly  in  the  study  of  fluids.  However,  a  further  study  of 


FIG.  47.  —  Two  Forces  not  in  Equilibrium. 

this  simple  case  can  hardly  fail  to  add  to  the  pupil's  understanding 
of  it.  For  this  purpose  we  may  use  two  spring  balances  and  a  board, 
the  latter  resting  upon  a  number  of  small  marbles  or  bicycle  balls 
placed  on  a  table  (Fig.  47).  Cords  are  attached  to  nails  at  A  and  B. 
Horizontal  forces  are  applied  to  the  board  through  these  cords,  and 
are  measured  by  the  balances.  If  these  forces  are  in  equilibrium  with 
each  other,  the  board  will  remain  at  rest;  if  they  are  not  in  equilib- 
rium, it  will  move,  since  the  friction  is  inappreciable.  The  experi- 
ment yields  the  following  results:  (i)  When  equal  forces  are  applied 
in  opposite  directions  but  not  along  the  same  line  (Fig.  47),  the  board 


70  STATICS  OF  SOLIDS 

will  not  be  in  equilibrium,  but  will  turn  round  until  the  forces  act 
along  the  same  line  (Fig.  48).  The  board  will  then  be  in  equilibrium. 
(2)  When  the  applied  forces  are  opposite  and  have  the  same  line  of 


FIG.  48. — Two  Forces  in  Equilibrium. 

action,  but  are  unequal,  the  board  will  be  pulled  in  the  direction  of 
the  greater  force.  (3)  When  the  forces  are  either  equal  or  unequal, 
but  not  opposite  in  direction,  the  board  will  be  moved.  Hence,  in 
general  — 

Two  forces  acting  upon  the  same  body  balance  each  other 
when  and  only  when  they  are  equal  in  magnitude,  opposite 
in  direction,  and  have  the  same  line  of  action. 

A  and  B  are  the  points  of  application  of  the  forces.  A  force  has 
the  same  effect  upon  a  solid  when  it 'is  applied  at  any  other  point  in 
the  same  line  of  action.  Thus,  if  either  of  the  equal  and  opposite 
forces  in  the  above  experiment  were  applied  at  C  (Fig.  48)  instead  of 
at  A  or  B,  the  two  would  still  be  in  equilibrium. 

67.  The  Elements  of  a  Force.  —  The  effect  of  a  force 
depends  upon  three  things,  namely,  its  magnitude,  its  direc- 
tion, and  its  line  of  action  (or  its  point  of  application) .  These 
are  called  the  elements  of  a  force.  They  must  all  be 
considered  in  describing  and  comparing  forces  and  in 
discussing  their  effects,  whether  the  forces  are  balanced 
or  unbalanced. 

•    68.   The  Geometrical  Representation  of  Forces.  —  The 

relative  magnitudes  and  directions  and  the  points  of  appli- 
cation of  a  set  of  forces  can  be  accurately  shown  in  a  dia- 
gram in  which  each  force  is  represented  by  a  straight  line. 
The  direction  of  the  force  is  represented  by  the  direction 


CONCURRENT    FORCES  71 

of  the  line,  as  indicated  by  an  arrow-head  placed  on  it; 
the  magnitude  of  the  force,  by  the  length  of  the  line; 
and  the  point  of  application  of  the  force,  by  the  point 
from  which  the  line  is  drawn  (the  end  of  the  line  from 
whrch  the  arrow-head  points)^ 

The  method  is  illustrated  in  Fig.  49,  which  represents 
two  forces  having  a  common  point  of  application,  0,  and 
differing  in  direction  by  a  right  angle. 
We  see  that  the  force  represented  by 
OB  is.  twice  as  great  as  the  other, 
since  the  line  representing  it  is  twice  o 


as  long;  but  the  diagram  does  not  give 
the  numerical  value  of  the  forces  unless  the  scale  adopted 
in  the  construction  is  known.  The  magnitude  of  a  force 
can  be  represented  on  any  scale  desired.  Thus  i  cm.  may 
represent  10  g.,  100  g.,  500  g.,  etc.;  but  all  forces  must  be 
represented  on  the  same  scale  in-  the  same  figure^ 

The  geometrical  representation  of  forces  and  their  relation  to  one 
another  is  exceedingly  useful,  and  is  constantly  employed  in  this  and 
the  following  chapters.  The  pupil  will  have  practise  in  the  con- 
struction and  use  of  such  diagrams  in  connection  with  the  laboratory 
work  and  in  the  solution  of  problems. 

69.  Equilibrium  of  Three  Concurrent  Forces.  —  Forces 
whose  lines  of  action  meet  in  a  point  are  called  concurrent 
forces.  Concurrent  forces  may  or  may  not  have  a  common 
point  of  application,  but,  if  not,  the  lines  along  which  they 
act  meet  in  a  point  when  produced. 

Experiment  shows  that  three  concurrent  forces  are  in 
equilibrium  only  when  certain  definite  relations,  as  regards 
magnitude  and  direction,  exist  among  them.  A  simple  form 
of  apparatus  for  studying  these  relations  is  shown  in  Fig. 
50.  Three  cords  are  tied  to  a  ring  and  a  spring  balance 
pulls  on  each.  Nails  or  clamps  are  provided  to  hold  the 


STATICS  OF  SOLIDS 


FIG.  50. 


balances  in  position  on  a  large  board  or  a  table.     The  ring 
moves  to  a  position  in  which  the  three  pulls  exerted  on  it 

are  in  equilibrium.  The  lines 
of  action  of  these  forces  lie  in 
the  same  plane  and  are  concur- 
rent at  or  near  the  center  of 
the  ring.  The  forces  act  out- 
ward from  this  point  in  the 
directions  of  the  three  cords, 
and  their  magnitudes  are  given 
by  the  readings  of  the  scales. 
In  order  to  determine  the  rela- 
tions which  exist  among  the 
forces,  they  are  represented  in  magnitude  and  direction 
by  the  lines  a,  b,  and  c,  respectively  (Fig.  51),  according 
to  the  rules  given  above.  If  the  experimental  work  and 
the  construction  are  accurate,  it  will  be 
found  that  the  diagonal  R  of  the  par- 
allelogram constructed  upon  any  two  of 
these  lines  as  sides  is  equal  to  the  third 
line,  and  is  in  exactly  the  opposite  direc- 
tion from  O.  The  conditions  necessary 
for  equilibrium  may  therefore  be  stated 
as  follows : 

Three  concurrent  forces,  acting 
on  the  same  body,  are  in  equi- 
librium only  when  their  lines  oj 
action  lie  in  the  same  plane 
and  their  magnitudes  and  direc- 
tions are  such  that,  if  the  lines  representing  any  two  of  them 
be  taken  as  the  sides  of  a  parallelogram,  the  concurrent 
diagonal  of  this  parallelogram  will  be  equal  and  opposite  to 
the  line  representing  the  third  force. 


FIG.  51.  —  Three  Concur- 
rent Forces,  a,  b,  and  c, 
in  Equilibrium. 


CONCURRENT    FORCES  73 

70.  Resultant  Force.  —  The  resultant  of  two  or  more 
forces,  acting  upon  the  same  body,  is  the  single  force 
that  would  produce  the  same  effect  upon  the  body  as  the 
given  forces,  if  it  were  substituted  for  them  (Art.  n).  It 
is  frequently  necessary  in  studying  mechanical  problems 
to  "find"  the  resultant  of  given  forces,  i.e.  to  determine 
the  magnitude,  direction,  and  line  of  action  of  the  single 
equivalent  force.  In  doing  this  we  are  not  in  the  least  con- 
cerned with  the  actual  or  possible  substitution  of  such  a 
force  for  the  given  forces.  Such  a  substitution  may  or  may 
not  be  possible. 

For  example,  if  a  cork  weighing  20  g.  is  released  under  water  and 
the  buoyant  force  of  the  water  upon  it  is  100  g.,  the  resultant  of  these 
forces  is  a  force  of  80  g.,  acting  vertically  upward;  by  which  we  mean 
that  the  cork  rises  through  the  water  just  as  rapidly  as  it  would  if 
it  were  acted  upon  by  a  single  upward  force  of  80  g.  in  place  of  its 
weight  and  buoyancy.  The  actual  impossibility  of  making  the 
substitution  in  this  case  does  not  enter  into  the  question. 

The  process  of  finding  the  resultant  of  two  or  more  given 
forces  is  called  the  composition  of  forces.  In  the  case  of 
forces  acting  in  both  directions  along  the  same  line,  the 
resultant  is  found  by  subtracting  the  sum  of  all  the  forces 
that  act  in  one  direction  along  the  line  from  the  sum  of 
all  the  forces  that  act  in  the  opposite  direction. 

For  example,  if  two  ropes  are  attached  to  opposite  ends  of  a  log  and 
two  boys  pull,  one  with  a  force  of  30  Ib.  and  the  other  with  a  force 
of  40  Ib.,  upon  one  rope  in  the  direction  of  the  length  of  the  log,  and 
a  third  boy  pulls  with  a  force  of  50  Ib.  in  the  opposite  direction  upon 
the  other  rope,  the  resultant  of  these  forces  will  be  a  force  of  20  Ib. 
in  the  direction  in  which  the  two  boys  pull.  Whether  the  log  will 
move  or  not  will  depend  upon  whether  a  force  of  20  Ib.  is  sufficient 
to  overcome  the  friction  between  the  log  and  the  ground. 

Other  methods  are  required  for  the  composition  of  forces  acting 
at  an  angle  with  one  another  or  along  different  parallel  lines,  as  is 
shown  in  the  following  articles. 


74  STATICS  OF  SOLIDS 

71.  Resultant  of  Two  Concurrent  Forces.  Parallelo- 
gram of  Forces.  —  Any  two  of  three  forces  in  equilibrium 
•may  be  regarded  as  together  balancing  the  third;  hence 
(by  definition)  their  resultant  is  the  single  force  which 
would  also  balance  the  third ;  hence,  further  (Art.  66) ,  this 
resultant  must  be  equal  and  opposite  to  the  third  and 
must  act  along  the  same  line. 

Referring  now  to  Fig.  51,  it  will  be  recalled  that  the 
^diagonal  R  is 'equal  and  opposite  to.  c;  hence  it  correctly 
represents  the  resultant  of  the  forces  denoted  by  a  and  b. 
The  sides  a  and  b  and  the  diagonal  R  of  the  parallelogram 
correctly  represent  the  relations  between  the  two  concur- 
rent forces  and  their  resultant  irrespective  of  any  third 
force ;  hence  — 

//  two  concurrent  forces  are  represented  by  lines  drawn  from 
the  same  point,  the  concurrent  diagonal  of  the  parallelogram 
constructed  upon  these  lines  as  sides  will  represent  their  result- 
ant in  magnitude  and  direction.  This  construction  is  known 
as  the  parallelogram  of  forces.  The  numerical  value  of  the 
resultant  is  found  by  accurately  measuring  the  length  of 
the  diagonal,  and  computing  the  force  that  this  length  of 
line  represents  according 'to  the  scale  adopted  in  the 
construction.  The  accuracy  of  the  numerical  result  will 
depend,  of  course,  upon  the  care  and  skill  exercised  in 
constructing  the  parallelogram  and  measuring  the  diagonal. 

The  resultanf  of  two  concurrent  forces  can  always  be  found  by 
the  parallelogram  construction.  It  can  also  be  computed  by  the  rules 
of  trigonometry.  In  a  few  special  cases  it  can  be  computed  from  the 
relations  established  in  plane  geometry.  The  most  important  of  these 
cases  is  that  of  two  forces  acting  at  an  angle  of  90°.  In  this  case 
the  resultant  is  equal  to  the  square  root  of  the  sum  of  the  squares 
of  the  given  forces  (since  the  square  of  the  hypothenuse  of  a  right 
triangle  is  equal  to  the  sum  of  the  squares  of  the  other  two  sides). 


CONCURRENT    FORCES  75 

72.  Composition  of  More  than  Two  Concurrent  Forces.  —  The 
resultant  of  any  number  of  concurrent  forces  can  be  found  by  combin- 
ing the  resultant  of  any  two  of  them  with  a  third,  their  resultant  with 
a  fourth,  and  so  on  till  each  force  has  been  included  once  in  the  con- 
struction or  computation.     The  last  resultant  is  the  resultant  of  all 
the  forces.     It  should  be  noted  that  we  have  the  privilege  of  com- 
bining the  forces  in  any  order;  for  it  sometimes  happens  that  the 
resultant  can  easily  be  computed  from  geometrical  relations  when 
a  certain  order  is  followed,  while  any  other  order  leads  to  difficulties. 
(See,  for  example,  the  eleventh  problem  in  the  following  set.) 

73.  Equilibrant.  —  The  single  force  that  would  balance 
one  or  more  given  forces  is  called  their  equilibrant.     The 
equilibrant  of  any  number  of  forces  is  equal  and  opposite 
to  their  resultant.    (Why?)    Either  of  two  forces  in  equilib- 
rium is  the  equilibrant  of  the  other;  and  any  one  of  three 
forces  in  equilibrium  is  the  equilibrant  of  the  other  two 
(Fig.  51)- 

74.  Resolution  of  a  Force.     Component  Forces.  —  It  is 

frequently  necessary  in  studying  the  effects  of  a  force  to 
consider  it  as  being  replaced  by  two  or  more  concurrent 
forces  which  are  together  equivalent  to  it.  The  process  of 
finding  the  required  set  of  equivalent  forces  is  the  reverse 
of  composition,  and  is  known  as  the  resolution  of  the  given 
force  into  its  components. 

It  is  effected  by  constructing  the  parallelogram  of  forces, 
as  will  readily  be  understood  by  referring  to  Fig.  51.  Since 
R  in  this  figure  denotes  a  force  which  is  equivalent  to  the 
forces  denoted  by  a  and  b,  it  follows  that  the  forces  denoted 
by  a  and  b  are  together  equivalent  to  the  force  denoted  by 
R.  The  following  example  will  serve  to  illustrate. 

A  block  of  wood  weighing  1200  g.  is  placed  on  a  plane,  AB 
(Fig.  5  2),  which  is  inclined  at  an  angle  such  that  the  length  of 
the  plane  AB  is  three  times  its  height  BC.  How  great  must 


76  STATICS  OF  SOLIDS 

friction  be  to  keep  the  block  from  sliding,  and  what  pressure  does 

the  block  exert  upon  the  plane? 
As  indicated  in  the  ques- 
tions, the  weight  of  the  block 
(represented   by  OW  in    the 
figure)  gives  rise  to  two  effects: 
(i)  a  tendency  of  the  block  to 
slide  down  the  plane,  and  (2) 
a  pressure  of  the  block  against 
the   plane.      The  first  effect 
would  be  produced  by  a  force 
FIG.  52.  —  Forces  OF  and  OP  are  together  of  a  certain  magnitude  acting 
equivalent  to  weight  OW.  on  the  block  in  the  direction  in 

which  it  tends  to  slide  (parallel  to  the  plane)  ;  .  and  the  second  effect 
would  be  produced  by  a  force  of  a  certain  magnitude  acting  on  the 
block  in  the  direction  in  which  it  presses  against  the  plane  (perpendic- 
ular to  the  plane).  The  problem  then  consists  in  finding  the  magni- 
tudes of  these  two  forces,  which  together  would  be  equivalent  to  the 
weight  of  the  block.  We  have,  therefore,  to  construct  the  parallelo- 
gram of  forces,  having  given  the  diagonal  OW  and  the  directions  OM 
and  ON  (but  not  the  lengths)  of  two  adjatent  sides.  The  sides  OF 
and  OP  of  this  parallelogram  represent,  therefore,  the  magnitudes  as 
well  as  the  directions  of  the  forces  sought.  Triangles  OFW  and  ABC 

are  similar.    (Why?)    Hence  7^.  =  -TR  =  "  5  from  which  OF 


3  3 

=  400  g.     If  a  force  of  400  g.  is  sufficient  to  overcome  friction,  the 

block  will  slide;  if  not,  it  will  remain  at  rest.     The  pressure  against 
the  plane,  represented  by  OP,  is  VI2O02  —  4OO2  =  1131.2  g. 

A  given  set  of  two  or  more  forces  has  only  one  resultant, 
as  is  shown  by  the  fact  that  only  one  parallelogram  can  be 
constructed  when  two  adjacent  sides  and  the  included  angle 
are  given;  but  any  number  of  different  sets  of  forces  can  be 
found  which  are  equivalent  to  a  given  force,  since  any  num- 
ber of  parallelograms  can  be  constructed  on  a  given  line 
as  a  diagonal.  When  there  are  only  two  components 
and  their  directions  are  given  or  determined,  there  is  but 
one  solution,  as  in  the  above  problem. 


CONCURRENT    FORCES 


77 


PROBLEMS 

Note.  —  The  following  problems  are  all  to  be  solved  by  computation, 
based  upon  known  geometrical  relations.  A  figure  drawn  with  a  rough 
approximation  to  accuracy,  as  in  geometrical  demonstrations,  will  serve  to 
present  these  relations  to  trie  eye. 

1.  A  weight  of  100  kg.  is  supported  by  two  cords  mak- 
ing equal  angles  with  the  horizontal  and  an  angle  of  120° 
with  each  other.    What  is  the  tension  on  each  cord? 

2.  What  would  be  the  tension  on  each  cord  supporting 
FIG.  53-         the  above  weight  if  one  made  an  angle  of  30°  with  the 

horizontal,  and  the  other  60°  ? 

3.  What  would  be  the  tension  on  each  cord  if  each  made  an  angle  of  45° 
with  the  horizontal? 

4.  A  ball  is  placed  on  a  plane  inclined  at  an  angle  of  30°.    What  fraction 
of  its  weight  tends  to  cause  motion  down  the  plane? 

5.  If  the  weight  of  the  ball  in  the  previous  problem  is  2  kg.,  what  pressure 
does  it  exert  upon  the  plane? 

6.  A  picture  weighing  20  Ib.  is  hung  by  a  cord 
passing  over  a  nail,  the  two  parts  of  the  cord  mak- 
ing an  angle  of  60°  with  each  other.  '  What  is  the 
tension  of  the  cord? 

7.  A  square  space  is  enclosed  by  passing  a  rope 
around  4  posts  at  the  corners,  and  the  tension  of 
the  rope  is  50  Ib.    What  is  the  magnitude  and  the 
direction  of  the  resultant  force  on  each  post? 

8.  The  beam  AB  (Fig.  54)  of  a  derrick  is  in- 
clined at  an  angle  of  30°  with  the  vertical  center 
post  AC,  and  a  weight  of  3  tons  hangs  from  the 

upper  end   of  the  beam.     Find  the   tension   of  the  horizontal  cable  BC. 

9.  A  hammock  is  suspended  from  two  hooks  at  the  same 
height  and  12   ft.   apart.     A   person  weighing   160  Ib.   sits 

B  at  the  center.  What  is  the  tension  of  the  ropes  (a)  if  the 
center  of  the  hammock  is  4  ft.  below  the  level  of  the  hooks? 
(6)  if  it  is  2.5  ft.  below? 

10.  A  street  lamp   weighing  80    Ib.   is  supported   by  a 
bracket  projecting  3   ft.  from  a  wall  (Fig.  55) .      The  brace 
AC  meets  the  wall  4  ft.  below  the  tie-rod,  AB.     Is  the  force 

sustained  by  the  tie-rod  a  pull  or  a  push?  the  force  sustained  by  the  brace? 
Compute  the  forces  sustained  by  each. 


FIG.  54. 


STATICS   OF  SOLIDS 


FIG.  56. 


11.  Three  ropes  pull  horizontally  upon  a  post  (Fig. 
56).    The  tension  of  A  is  50  lb.,  of  B  i6o.'lb.,  and  of  C 
nolb.     A  and  B  pull  in  the  same  straight  line,  and  C 
at  right  angles  to-them.     Find  the  resultant  force  on  the 
post. 

12.  As  the  angle  between  two  forces  increases  from 
o°  to  180°,  how  does  their  resultant  vary?    What  is  the 
value  of  the  resultant  at  the  beginning?  at  the  end? 


II.     PARALLEL  FORCES 

75.  Equilibrium  of  Three  Parallel  Forces.  —  Parallel 
forces  are  forces  having  parallel  lines  of  action.  It  is  found 
by  experiment  that,  if  three  parallel  forces  acting  on  the 
same  body  are  in  equilib- 
rium, the  following  condi- 
tions are  always  fulfilled: 

1.  The  three  forces,  f\,  /2, 
and  /3  (Fig.  57),  are  in  one 
plane. 

2.  The  two  outside  forces 
act  in  the  same  direction,  and 
the  inside  force  in  the  oppo- 
site direction. 

3 .  The  inside  force  is  equal 
to  the  sum  of  the  other  two. 

4.  The  outside  forces  are 

inversely  proportional  to  the  distances  (i.e.  the  shortest,  or  per- 
pendicular distances)  of  their  lines  of  action  from  the  line  of 
action  of  the  inside  force;  i.e.  f\  :/2  ::  d2  :  di}  or/i^i  =f^d^ 
It  should  be  noted  that  the  inside  force  is  nearer  the 
larger  of  the  outside  forces  if  they  are  unequal;  if  they 
are  equal,  it  is  midway  between  them.  The  points  of 
application  of  the  forces  need  not  lie  in  a  straight  line  (Fig. 
58).  Any  one  of  the  three  forces  may  be  regarded  as  the 
equilibrant  of  the  other  two. 


FIG.  57.— Three  Parallel  Forces  in 
Equilibrium. 


PARALLEL    FORCES 


79 


76.  Resultant  of  Two  Parallel  Forces  Having  the  Same 
Direction. — When  three  parallel  forces  are  in  equilibrium, 
the  two  outside  forces  together  balance  the  third  force; 
hence  their  resultant  would  also  balance  it.  This  result- 


FIG.  58.  —  General  Case  of  Three  Par-      FIG.  59.  —  Resultant  of  Two  Parallel 
allel  Forces  in  Equilibrium.  Forces,  f\  and  /2. 

ant, /3  (Fig.  59),  must,  therefore,  have  the  same  line  of 
action  as  the  third  force,  and  must  be  equal  to  it  in  magni- 
tude and  opposite  in  direction ;  hence  — 

The  resultant  of  two  parallel  forces  having  the  same  direc- 
tion is  equal  to  their  sum,  it  acts  in  the  same  direction  as  the 
component  forces,  and  its  line  of  action  divides  the  distance 
between  them  into  parts  which  are  inversely  proportional  to 

the  forces. 

PROBLEMS 

1.  ?Two  boys,  A  and  B,  carry  a  load  between  them  suspended  from  a 
pole  5  ft.  long.      The  load  is  2  ft.  from  ^4's  end.      What  fraction  of  it 
does  A  carry? 

2.  If  the  load  weighs  161  lb.,  where  must  it 
be  hung  in  order  that  A  may  carry  Q2lb.-of  it? 

3.  Two  horses   draw  a  plow  by  means  of  a 
doubletree,  D  (Fig.  60),  at  each  end   of  which 
is    a    singletree,    S.       What    are    the    relative 
magnitudes   and  directions   of    the   three   forces 
acting  on  the  doubletree,  if  its  arms  are  of  equal 
length?    If  the   dcyubletree   is  4    ft.    long,   what 

must  be  the  length  of  each  arm  in  order  that  one  of  the  horses  shall  draw 
three  fifths  of  the  load? 


FIG.  60. 


8o 


STATICS   OF   SOLIDS 


4.  A  man  carries  a  weight  of  20  Ib.  on  the  end  of  a  stick  3  ft.  long, 
placed  over  his  shoulder.  He  holds  the  stick  at  the  other  end.  What  is 
the  pressure  on  his  shoulder  (a)  if  the  distance  from  his  shoulder  to  his 
hand  is  2  ft.?  (6)  if  this  distance  is  i  ft.? 

III.     MOMENTS  or  FORCE 

77.  The  Rotative  Action  of  a  Force.  —  The  conditions 
which  determine  the  greater  or  less  effect  of  a  force  in  pro- 
ducing or  opposing  rotation  are  well  illustrated  in  the 
child's  game  of  seesaw. 


FIG.  61.  —  Moments  of  Force  in  Equilibrium,  fidi  —  facto. 


Two  boys  of  equal  weight  will  sit  at  equal  distances  from  the  axis 
about  which  the  board  turns;  but  if  they  are  of  unequal  weight,  the 
lighter  boy  will  sit  at  the  greater  distance.  The  boy  at  the  top  makes 
his  end  descend  by  leaning  backward,  thus,  in  effect,  increasing  his 
distance  from  the  axis.  The  boy  at  the  bottom  decreases  his  distance 
by  leaning  forward.  Evidently  the  effectiveness  of  a  force  (in  this 
case  the  weight  of  the  boy)  in  producing  rotation  depends  in  part 
upon  the  magnitude  of  the  force  and  in  part  upon  its  distance  from 
the  axis.  The  exact  relation  is  easily  derived  from  an  experiment 
patterned  after  the  seesaw.  A  slender  stick  (meter  rod)  is  supported 
on  a  horizontal  axis  (a  nail),  through  a  hole  so  situated  that  the  rod 
comes  to  rest  in  a  horizontal  position.  Two  weights,  either  equal  or 
unequal,  hung  from  the  rod  on  opposite  sides  of  the  axis  can  be  so 
adjusted  as  to  leave  the  rod  in  equilibrium  (Fig.  61).  With  this 
adjustment,  the  tendency  of  the  one  weight  to  pull  its  end  of  the  rod 
down  is  balanced  by  the  equal  and  opposite  tendency  of  the  other 
weight  to  pull  its  end  down.  In  every  such  case  the  weights  are 


MOMENTS    OF    FORCE  81 

inversely  proportional  to  their  distances  from  the  axis  (f\  :/2"  02  :  fli), 
or  the  product  of  one  force  and  its  distance  from  the  axis  is  equal 
to  the  product  of  the  other  force  and  its  distance. 
If  one  of  the  weights  is  replaced  by  a  spring 


balance,  a  measured  pull  can  be  exerted  in    i  .  °,''    \    i 

any  direction  (Fig.  62).     It  will  then  be  found  j\ 

that  the  force  fa  required  to  maintain  equi-    £_\/i  \  2 

librium  increases  as  the  direction  of  the  force  FlG  62 

changes  from  the  vertical  toward  the  horizon- 

tal. But,  at  the  same  time,  the  distance  02  from  the  line  of  action  of 
the  force  to  the  axis  decreases,  so  that  the  product  faa^  remains  equal 
to  the  product  f\a\.  The  products  f\a\  and  /2#2,  therefore,  measure 
the  effectiveness  of  the  forces  in  producing  or  opposing  rotation. 

The  effectiveness  of  a  force  in  producing  or  opposing 
rotation  about  a  given  axis  is  called  the  moment  of  the  force 
with  respect  to  that  axis.  The  moment  of  a  force  is  meas- 
ured by  the  product  of  the  force  and  the  distance  (i.e.  the 
perpendicular  distance)  of  its  line  of  action  from  the  axis. 
This  perpendicular  distance  is  called  the  arm  of  the  force. 
Thus  in  the  above  experiment  ai  is  the  arm  of  the  force  /i 
with  respect  to  an  axis  at  0,  and  /i#i  is  the  moment  of  the 
force;  a*  is  the  arm  of  the  force  /2,  and/2a2  is  its  moment 
with  respect  to  the  same  axis. 

78.  Equilibrium  of  Moments.  —  The  general  condition 
necessary  for  the  equilibrium  of  a  body  with  respect  to 
rotation  is  that  the  sum  of  all  the  moments  of  force  tend- 
ing to  turn  the  body  in  one  direction  round 
any  axis  must  be  equal  to  the  sum  of  all 
the  moments  of  force  tending  to  turn  the 
body  in  the  opposite  direction  round  the 
same  axis. 

Direction  round  an  axis  is  termed 
clockwise,  if  it  is  as  the  hands  of  a  clock  turn,  and  counter- 
clockwise, if  in  the  opposite  direction  round.  A  clockwise 


82  STATICS  OF  SOLIDS 

and   a   counter-clockwise  moment  may  act  on  the  same 
side  of  an  axis  and  be  in  equilibrium  (Fig.  63). 

In  the  case  shown  in  Fig.  64  there  are  two  clockwise  moments, 
and  /3«3,  and  one  counter-clockwise  moment,  f\a\.  Hence  the 
condition  for  equilibrium  with  respect  to,  an 
axis  at  O  is/i#i  =/202  +  faaz-  The  force  repre- 
sented by  /4  does  not  tend  to  cause  rotation 
in  either  direction,  since  its  line  of  action 
passes  through  the  axis.  Its  arm  is  zero  and 
its  moment  is  zero.  The  same  is  true  of  the 
pressure  at  the  axis. 

The  law  as  stated  above  holds  for  any 
body,  whether  actually  supported  on  an  axis  or 
not.  An  axis  may  be  assumed  in  any  position, 
and,  if  the  body  is  in  equilibrium,  the  sum  of  the  clockwise  moments, 
taken  with  respect  to  the  assumed  axis,  will  be  equal  to  the  sum  of  the 
counter-clockwise  moments,  taken  with  respect  to  the  same  axis. 
Thus  in  the  case  of  equilibrium  shown  in  Fig.  58,  assuming  an  axis 
at  the  point  of  application  of  /s,  the  moment  of  this  force  is  zero,  the 
moment  of  /i  is/i^i,  and  the  moment  of  /2  isfodz.  But  in  the  previous 
study  of  this  case  we  learned  that/idi  =  fzdz;  and  since  these  moments 
are  opposite  in  direction,  they  fulfil  the  conditions  necessary  for 
equilibrium.  Again,  if  the  axis  is  assumed  at  the  point  of  appli- 
cation of  /i,  the  moment  of  this  force  is  now  zero,  the  moment  of  fa 
isfadi,  and  the  moment  of  /2  is/2  (^i  +  d%). 
But/2(</i  +  4)  =M  +M, 

=  fadi  +fidi,        (since  fidi 


=  fadi,  (since  /3  =  /i 

That  is,  the  moments  are  equal  about  this  axis  also.  Similarly, 
it  can  be  shown  that  the  moments  of  these  three  forces  are  in 
equilibrium  with  respect  to  an  axis  taken  in  any  position. 

79.  Mechanical  Couple.  —  Two  equal  forces  acting  in 
opposite  directions  upon  the  same  body  and  having  differ- 
ent lines  of  action  constitute  a  couple  (Fig.  65).  The  dis- 
tance between  the  lines  of  action  is  called  the  arm  of 
the  couple.  The  moment  of  a  couple  about  any  axis 


MOMENTS    OF    FORCE  83 

perpendicular  to  the  plane  of  the  couple  is  the  product  of 
the  arm  and  either  force  (  =  fa  in  the  figure).  (Prove  that 
this  is  so  for  an  axis  at  O,  midway  between 
the  points  of  application  of  the  forces,  and  also 
for  an  axis  at  Q,  a  point  anywhere  in  the  plane.) 
A  couple  has  neither  a  single  equilibrant  nor 
resultant;  but  if  the  body  upon  which  it  acts 
is  supported  on  a  fixed  axis,  the  moment  of 
the  couple  can  be  balanced  by  a  single  oppos- 
ing moment.  The  rolling  of  a  ship  is  due  to  the  action  of 
a  couple,  consisting  of  the  weight  of  the  ship  and  the 
buoyant  force  of  the  water  (Art.  85). 

PROBLEMS 

1.   If  in  Fig.  6 1  f\  =  90  g.,  a\  —  40  cm.,  and  02  =  25  cm.,  what  is  the 
value  of  fz 


FIG.  65. 


FIG.  66.  FIG.  67. 

2.  If  in  the  same  figure /i  =  250  g.,/2  =  125  g.,  and  az  =  50  cm.,  what  is 
the  value  of  ai? 

3.  An  object  weighed  with  a  steelyard  (Fig.  66)  is  how  many  times 
heavier  than  the  weight  upon  the  beam  by  which  it  is  balanced? 

4.  A  man  in  moving  a  stone  with  a  crowbar  exerts  a  force  of  50  Ib.  with 

each  hand  at  distances  of  100  cm.  and  150  cm.,  respectively, 
from  the  point  where  the  crowbar  is  supported,  and  this 
point  of  support  is  25  cm.  from  the  lower  end  of  the  bar. 
How  great  is  the  force  exerted  at  this  end? 

6.  In  drawing  a  nail  with  a  claw-hammer  a  man  exerts  a 
force  of  50  Ib.  The  point  where  the  hammer  presses  against 
the  board  is  i  in.  from  the  nail  and  9  in.  from  the  point  on 


FIG.  68. 


the  handle  where  the  force  is  applied.     How  great  is  the  pull  upon  the  nail? 


84  STATICS  OF  SOLIDS 

6.  If  the  arms  of  a  balance  (Fig.  4)  are  not  of  exactly  equal  length, 
what  error  in  weighing  results  when  the  body  to  be  weighed  is  placed  in 
the  pan  on  the  longer  arm? 

7.  The  safety  valve  V  (Fig.  69)  covers  a  boiler  opening  whose  area  is 

.4  sq.  in.  The  distance  AC  is  1.5 
in.,  and  the  ball  P  weighs  12  lb. 
*^  Disregarding  the  weight  of  the  valve 
and  lever,  what  must  be  the  dis- 
tance AB  in  order  that  the  valve 
shall  open  under  a  steam  pressure 


FIG.  69.  —  Safety  Valve.  of  X5°  lb-  to  the  square  inch  ? 

8.  A  stiff  pole  10  ft.  long  pro- 
jects horizontally  from  a  vertical  wall.  It  is  just  strong  enough  to  sustain 
a  weight  of  30  lb.,  hung  at  the  outer  end.  How  far  out  on  the  pole  may 
a  boy  weighing  90  lb.  venture  with  safety? 

IV.  EFFECT  OF  WEIGHT  ON  THE  EQUILIBRIUM  OF  BODIES 

80.  Gravity  and  the  Center  of  Gravity.  —  The  earth's 
attraction  for  bodies  is  called  gravity,  when  named  in  a 
general  way  without  reference  to  its  amount  for  any  par- 
ticular body.  Gravity  acts  on  every  particle  of  a  body, 
and  these  forces  on  the  individual  particles  are  all  directed 
toward  the  same  point  at  (or  near)  the  earth's  center. 
Since  this  point  is  at  a  distance  of  about  four  thousand  miles, 
there  is  no  measurable  error  in  assuming  that  the  forces 
of  gravity  acting  on  the  different  parts  of  a  body  are  par- 
allel. Their  resultant  is  therefore  equal  to  their  sum,  and 
is  what  we  know  as  the  weight  of  the  body. 

The  direction  of  this  resultant,  like  that  of  its  compo- 
nents, is  vertical;  its  point  of  application  can  be  shown  both 
experimentally  and  mathematically  to  be  a  fixed  point 
relative  to  the  body,  however  the  body  may  be  situated 
and  in  whatever  condition  of  rest  or  of  motion  it  may  be, 
provided  it  retains  the  same  shape  and  the  same  distri- 
bution of  its  mass  throughout  its  bulk.  In  other  words, 


EFFECT    OF    WEIGHT    ON    EQUILIBRIUM  85 

a  rigid  body  behaves  as  if  the  earths  attraction  for  it  were  a 
single  vertical  force,  acting  always  and  in  all  circumstances 
at  one  and  the  same  point  of  the  body.  This  point  is  called 
the  center  of  gravity  of  the  body.  Other  names  for  it  are 
center  of  weight,  center  of  mass,  and  center  of  inertia. 

The  direction  of  gravity  is  given  by  a  plumb-line,  which 
is  a  cord  from  which  a  weight  hangs  at  rest. 

81.   Effect  of  Weight  on  a  Suspended    Body.  —  If  a 

body  is  free  to  rotate  about  an  axis  from  which  it  is  sus- 
pended, it  always  comes  to  rest  with  its  center  of  gravity 
vertically  below  the  axis. 

The  reason  for  this  behavior  is  shown  in  Fig.  70,  which 
represents  a  flat  body,  as  a  board,  suspended  at  0.  The 
weight  of  the  body,  w,  is  regarded  as  a  single  force  applied 
at  the  center  of  gravity,  C.  The  body 
is  evidently  not  in  equilibrium  in  the 
position  represented  in  the  figure,  since 
the  moment  of  its  weight,  wa,  is  un- 
balanced and  causes  rotation.  When  C 
is  vertically  below  O,  the  moment  of  FlG" 7a 

the  weight  of  the  body  is  zero,  since  its  arm  is  then  zero. 
In  this  position  the  weight  of  the  body  and  the  pressure 
upon  it  at. the  axis  are  equal  and  opposite 
and  have  the  same  line  of  action.  This  is 
therefore  a  position  of  equilibrium. 


82.    Methods  of  Finding  the  Center  of  Gravity 
of  a  Body.  —  The  behavior  of  the  center  of  grav- 
ity of  a  body  when  suspended   affords  a  simple 
method  of  finding  it  experimentally,  as  illustrated  in 
FIG.  71.  Fig.  71.     The  figure  represents  any  flat  body,  as  a 

board,  suspended  at  0  upon  a  pin  or  a  nail. 
The  vertical  through  O  is  determined  by  a  small  plumb-line  sus- 
pended from  the  same  axis.     A  line  indicating  the  position  of  this 


86 


STATICS  OF  SOLIDS 


vertical  is  drawn  on  the  body.  The  center  of  gravity  of  the  body 
is  at  some  point  within  it,  directly  back  of  this  line.  The  body  is 
then  suspended  at  some  other  point,  P,  and  a  second  vertical,  DP, 
determined  as  before.  Since  the  center  of  gravity  lies  directly  back 
of  both  verticals,  it  must  be  the  one  point  that  lies  directly  back  of 
their  intersection,  midway  between  the  front  and  back  surfaces. 

The  center  of  gravity  of  any  body  may  be  similarly  found  by 
suspending  it  successively  at  any  two  points  about  which  it  is 
free  to  swing  (Fig.  72).  The  only  difficulty,  if  any,  would  be  in 
accurately  determining  the  verticals  through  the  substance  of  the 
body. 

The  center  of  gravity  of  a  body  of  regular  shape  and  uniform  density 
is  its  center  of  figure.  The  sphere,  spheroid,  cube,  parallelepiped, 
and  circular  cylinder  are  examples.  The  shape  of  a  body  may  be 

such  that  its  center  of  gravity 
does  not  lie  in  any  material  part 
of  it;  e.g.  a  ring  or  a  hollow 
sphere. 


FIG.  72.  —  Finding  the  Center  of 
Gravity. 


83.  States  or  Kinds  of 
Equilibrium.  —  A  body  at 
rest  may  or  may  not  be  in 
equilibrium.  It  is  in  equi- 
librium provided  there  is  no 
unbalanced  force  or  moment 
of  force  acting  upon  it  at  the  instant.  A  body  thrown  ver- 
tically upward  is  at  rest  at  the  instant  when  it  ceases  to 
rise;  but  it  is  not  in  equilibrium,  since  its  weight  is  then 
unbalanced,  just  as  it  is  while  the  body  is  rising  or  falling. 
So,  too,  a  swing  or  a  pendulum  is  at  rest  but  not  in  equi- 
librium at  the  end  of  each  vibration,  for  the  moment  of 
its  weight  is  unbalanced.  On  the  other  hand,  a  body  at 
rest  may  be  in  equilibrium,  although  it  does  not  continue 
at  rest;  for,  under  certain  conditions,  equilibrium  is  very 
unstable  or  insecure  and  is  maintained  only  by  the  exercise 
of  considerable  skill.  The  feats  of  balancing  performed 


EFFECT    OF    WEIGHT    ON    EQUILIBRIUM  87 

by  acrobats  are  of  this  class.  (Fig.  73.)  Equilibrium  is 
of  three  kinds  or  states,  viz.,  stable,  unstable,  and  neu- 
tral. The  equilibrium  of  a 
body  in  a  given  position  is 
stable  if  the  body  tends  to 
return  to  that  position  after 
being  turned  or  tilted  very 
slightly  in  any  direction;  it  is 
unstable  if  the  body  tends  to 
move  still  farther  from  its 
original  position  after  being 

thus    disturbed;    it    is     neutral      FIG.  73.  — Unstable  Equilibrium. 

if  the  body  remains  in  equilibrium  in  any  adjacent  position. 
A  body  suspended  at  a  point  other  than 
its  center  of  gravity  is  in  stable  equilibrium 
when  at  rest  with  its  center  of 
gravity  vertically  below  the  sup- 
port; for  the  unbalanced  moment 
of  its  weight  turns  it  back  to  this 
position  when  it  is  displaced  in 
either  direction  (Figs.  70  and  71). 
A  rectangular  block  standing  on  a  table  is  in  stable  equi- 
librium (Fig.  74).     When  the  block  is  tilted  slightly,  the 
moment  of  its  weight,  acting  about  the  edge  on  which  it 
is  turned,  tends  to  bring  it  back  to  its  former  position. 
When  any  object  is  turned  or  tilted  out  of  a  position  of 
stable  equilibrium,  its  center  of  gravity  is  raised;  and  in 
returning  to  that  position,  the  center  of  gravity  falls. 

A  body  is  in  unstable  equilibrium  when  balanced  on  a 
point,  an  edge  (Fig.  75),  or  an  axis  with  its  center  of  gravity 
vertically  above  the  support.  When  any  object  is  in  a 
position  of  unstable  equilibrium,  the  moment  of  its  weight, 
taken  about  the  support  as  an  axis,  is  zero;  but  when 


iio 
FIG.  74- 


FIG.  75- 


88 


STATICS   OF   SOLIDS 


the  body  is  turned  or  tilted  ever  so  slightly,  the  moment 
of  its  weight  begins  to  act  in  the  direction  of  the  motion. 
Hence  the  slightest  disturbance  causes  the  body  to  fall. 
When  a  body  is  in  unstable  equilibrium,  its  center  of  grav- 
ity is  at  the  highest  possible  point  for  the  given  support. 

A  body  supported  on  an  axis  passing  through  its  center 
of  gravity  is  in  neutral  equilibrium  when  at  rest  in  any  posi- 
tion, since  the  moment  of  its  weight  is  zero  for  all  positions 
of  the  body  about  the  axis.  When  a  body  thus  supported 
is  set  rotating,  it  will  come  to  rest  in  any  position  in  which 
friction  may  chance  to  stop  it.  A  perfectly  balanced  wheel 
is  a  familiar  example.  The  equilibrium  of  a  sphere  on  a 
horizontal  plane  is  neutral,  since  its  center  of  gravity  re- 
mains vertically  above  the  point  of  support,  in  whatever 
direction  the  sphere  is  turned.  The  center  of  gravity  of  a 
body  is  neither  raised  nor  lowered  by  any  slight  displace- 
ment from  a  position  of  neutral  equilibrium. 

A  body  may  be  in  different  states  of  equilibrium  at  the  same  time, 
with  respect  to  motion  in  different  directions.  Thus  a  circular 
cylinder,  at  rest  on  its  side  upon  a  horizontal  surface,  is  in  neutral 
equilibrium  with  respect  to  rolling,  and  in  stable  equilibrium  with 
respect  to  the  upward  turning  of  either  end. 

84.  Stability.  —  A  body  in  stable  equilibrium  has  greater 
or  less  stability  according  as  its  weight  offers  greater  or 
less  opposition  when  an  attempt  is  made  to  overturn  it. 

Other  conditions  remaining 
the  same,  the  stability  of  a 
body  is  greater  the  greater  its 
weight,  the  larger  the  area 
of  its  base,  or  the  lower  its 
center  of  gravity. 


FIG.  76. 


The  effect  of  a  large  base  can  be 
shown  with  a  long  block  of  wood, 


EFFECT    OF    WEIGHT    ON    EQUILIBRIUM  89 

to  which  a  thin  board  is  nailed  at  one  end  (Fig.  76).  When  it  stands 
on  the  larger  base,  the  resisting  moment  of  force,  wa,  is  much 
greater,  and  the  body  must  be  turned  much  farther  against  this 
moment  before  it  will  fall  over.  (Show  this  from  the  figure.)  The 
effect  of  a  low  center  of  gravity  can  be  shown  with  a  block  weighted 
with  lead  at  one  end  (Fig.  77).  When  standing  on  its  heavier  end, 
the  block  must  be  turned  farther  before  its  weight  acts  to  overturn  it. 


FIG  77- 


FIG.  78.  — a,  Stable  Equilibrium;    b,   Non-Equilib- 
rium; c,  Unstable  Equilibrium. 


Tall  buildings,  high  towers  and  monuments,  etc.,  must  be  able  to 
withstand  an  enormous  overturning  moment  due  to  wind  pressure. 
A  maximum  wind  pressure  of  30  Ib.  per  sq.  ft.  from  top  to  bottom  of 
steel-frame  buildings  is  provided  for  in  the  Building  Code  of  New 
York  City,  and  this  must  not  be  greater  than  75%  'of  the  stability 
moment  of  the  structure. 

85.  Equilibrium  and  Stability  of  Floating  Bodies.  —  The 
buoyant  force  upon  a  floating  body  is  a  resultant  force 
equal  to  the  weight  of  the  body  (Art.  38).  Its  direction  is 
always  vertically  upward,  and  its  point  of  application  is 
the  center  of  gravity  of  the  displaced  liquid.  This  point 
is  called  the  center  of  buoyancy.  The  behavior  of  a  float- 
ing body  is  therefore  determined  by  the  two  resultant  forces, 
weight  and  buoyancy,  which  are  always  equal  and  oppo- 
site in  direction.  The  body  is  in  equilibrium  when  these 
forces  have  the  same  line  of  action,  i.e.  when  the  center 
of  gravity  of  the  body  and  the  center  of  buoyancy  lie  in 
the  same  vertical  line,  as  in  a  and  c  of  Fig.  78. 

C  denotes  the  center  of  gravity  of  the  body  (a  block  of  wood), 
and  B  the  center  of  buoyancy.  (The  arrows  are  directed  toward  the 
points  of  application  of  the  forces.) 


go  STATICS  OF  SOLIDS 

When  a  floating  body  is  inclined  from  a  position  of  equilibrium, 
the  center  of  buoyancy  shifts  toward  the  deeper  displacement,  as 
shown  in  b  of  the  figure;  and  the  lines  of  action  of  the  weight  of  the 
body  and  the  buoyant  force  no  longer  coincide.  The  two  forces 
then  act  as  a  couple.  If  the  body  was  originally  in  stable  equilibrium 
as  in  a,  the  couple  opposes  the  tilting  and  turns  the  body  back  again; 
if  the  equilibrium  was  unstable,  as  in  c,  the  couple  acts  in  the  direc- 
tion of  the  motion,  and  the  body  overturns. 

The  equilibrium  of  a  floating  body  is  always  stable  when  its  center 
of  gravity  is  below  the  center  of  buoyancy;  when  it  is  above  the 
center  of  buoyancy,  the  equilibrium  may  be  either  stable  or  unstable, 
as  shown  in  the  figure.  The  equilibrium  is  neutral  if  the  centers  of 
gravity  and  buoyancy  remain  in  the  same  vertical  line  when  the 
body  rolls  about  in  the  water,  as  is  the  case  with  a  sphere  or  a  long 
cylinder. 

The  stability  of  a  floating  body  of  given  shape  is  increased  by 
lowering  its  center  of  gravity;  for  this  increases  the  couple  which 
opposes  overturning.  (Draw  figure  to  illustrate.)  The  heavy 
engines  and  boilers  of  a  steamship  add  greatly  to  its  stability,  as  they 
are  placed  low  in  the  hull.  A  sailing  vessel,  on  a  voyage  without  a 
cargo,  carries  ballast  of  sand  or  other  waste  material  placed  in  the 
hold.  Without  cargo  or  ballast,  the  vessel  would  be  in  constant 
danger  of  overturning. 

PROBLEMS 

1.  In  what  direction  does  a  person  lean  when  carrying  a  heavy  load  in 
one  hand?    Why? 

2.  Show  that  when  a  homogeneous  hemisphere  is  inclined  (A, 
Fig.  79),  its  weight  tends  to  bring  it  into  the  position  shown  in  B. 
In  what  kind  of  equilibrium- is  it  in  the  second  position?     Is  it  in 
unstable  equilibrium  in  the  first  position?     Give  reasons. 


B 

FIG.  79.  FIG.  80.  FIG.  81. 

3.  (a)  Oil  cans  are  made  of  the  shape  shown  in  Fig.  80,  and  are  weighted 
with  lead  at  the  bottom.  Such  a  can  rights  itself  when  tipped.  Explain. 
(b)  Does  the  can  really  rise  or  fall  when  it  rights  itself? 


EFFECT    OF    WEIGHT    ON    EQUILIBRIUM  91 

4.  Why  does  a  person  always  lean  forward  before  attempting  to  rise 
from  a  chair? 

5.  A  pencil  with  a  knife  attached  can  be  balanced,  as  shown  in  Fig.  81. 
Try  it.     What  is  the  evidence  that  the  equilibrium  is  stable?     Where  is  the 
center  of  gravity  of  the  pencil  and  knife  regarded  as  one  body? 

6.  Show  by  means  of  figures  that  the  moment  of  the  weight  of  a  sphere 
is  zero  upon  a  horizontal  surface,  but  not  upon  an  inclined  plane. 

7.  If  a  body  that  will  not  roll  remains  at  rest  when  placed  on  an  inclined 
plane,  three  forces  act  to  hold  it  in  equilibrium.     Two  of  these  forces  are  its 
weight  and  the  pressure  of  the  plane.     What  is  the  third  force,  and  in  what 
direction  does  it  act?     Draw  a  figure  correctly  representing  the  direction 
and  the  relative  magnitude  of  the  three  forces. 

8.  Two  spheres  weighing  50  kg.  and  15  kg.,  respectively,  are  connected 
by  a  rod  so  that  the  distance  between  their  centers  is  80  cm.     Disregarding 
the  weight  of  the  rod,  where  is  the  center  of  gravity  of  the  whole  considered 
as  one  body? 

9  The  average  distance  between  the  centers  of  the  earth  and  the  moon 
is  about  240,000  miles;  the  mass  of  the  earth  is  80  times  that  of  the  moon. 
How  far  is  their  common  center  of  gravity  from  the  earth's  center? 

10.  Two  men,  A  and  B,  carry  a  board  30  ft.  long  and  of  -uniform  cross- 
section.     A  holds  at  one  end;  where  must  B  hold  in  order  to  carry  .6  of  the 
load? 

11.  A  boy  weighing  40  Ib.  wishes  to  seesaw  alone  on  a  plank  weighing 
70  Ib.     The  plank  is  24  ft.  long,  and  the  center  of  gravity  of  the  boy  is  i  ft. 
from  an  end  of  the  plank.     How  far  from  that  end  must  the  plank  be  sup- 
ported? 

12.  Why  is  it  an  advantage  to  spread  the  feet  when  standing  upon  a 
surface  that  is  moving  unsteadily,  as  the  deck  of  a  vessel? 

13.  What  would  happen  to  the  leaning  tower  of  Pisa  (Fig.  115)  if  the 
vertical  through  its  center  of  gravity  fell  without  the  base  of  the  tower? 

14.  Is  the  stability  of  a  boat  greater  when  the  occupants  are  standing 
or  sitting?    Why? 

15.  Why  is  it  difficult  to  walk  on  stilts? 

16.  A  uniform  stick  of  timber  10  ft.  long  balances 
on  an  axis  3  ft.  from  one  end  when  a  weight  of  20  Ib. 
is  hung  from  that  end.     Find  the  weight  of  the  stick. 

17.  Why  cannot  one  stand  with  his  heels  against  a 
wall  and  lean  forward  without  falling? 

18.  Two  boys,  A  and  B,  carry  a  uniform  plank  24  ft.  long,  weighing 
120  Ib.     A  holds  at  one  end  and  B  4  ft.  from  the  other  end.     What  load 
does  each  carry? 


92  STATICS  OF  SOLIDS 

19.  A  uniform  stick  10  ft.  long  and  weighing  20  Ib.  is  suspended  at  one 
end,  and  drawn  out  45°  from  the  vertical  by  a  force  /  (Fig.  82),  applied  at  the 
lower  end  and  at  right  angles  to  the  length  of  the  stick.    Find  the  value  of/. 

20.  Does  a  plane  through  the  center  of  gravity  of  a  body  always  divide 
the  body  into  parts  having  equal  weight?     Discuss  various  special  cases 
and  try  to  arrive  at  some  general  conclusions. 

V.  ELASTICITY.    STRESSES  AND  STRAINS 

86.  Effects  of  Balanced  Forces.  —  While  balanced  forces 
neutralize  each  other  so  far  as  the  motion  of  bodies  is  con- 
cerned, their  united  action  produces  changes  of  size  and 
shape,  and  other  effects  which  are  often  very  important. 
Under  the  action  of  balanced  pushes  and  pulls  solids  are 
compressed,  stretched,  bent,  twisted,  and  broken;  and  these 
effects  are  accompanied  by  internal  changes,  such  as  the 
crowding  together  of  the  particles  of  a  piece  of  wood  when 
pressed  between  the  jaws  of  a  vise,  and  the  slipping  of  the 
particles  of  a  piece  of  lead  over  one  another  when  the  lead 
is  hammered. 

Only  a  perfectly  rigid  body,  i.e.  one  whose  parts  can  not  be 
moved  relatively  to  one  another,  would  be  free  from  these 
effects,  and  experience  teaches  that  no  such  bodies  exist. 
Tempered  steel  is  one  of  the  most  rigid  of  substances;  yet 
fine  steel  wire  is  very  flexible,  and  even  such  thick  masses 
as  steel  rails  can  be  bent  considerably  without  breaking. 

A  knowledge  of  the  distribution  of  forces  throughout 
the  parts  of  bridges,  buildings,  and  other  structures,  and 
of  the  strength  of  building  materials  to  withstand  these 
forces  is  of  the  greatest  importance  to  engineers  and  archi- 
tects. Our  present  study  of  balanced  forces  will  afford  at 
least  a  partial  acquaintance  with  some  of  these  interesting 
problems. 

87.  Strains  and  Stresses.  —  Any  change  in  the  shape 
or  size  of  a  body,  due  to  the  action  of  force,  is  called  a 


STRESSES    AND    STRAINS  93 

strain.  A  body  in  a  state  of  strain  offers  resistance  to  the 
applied  forces  which  produce  the  strain.  Thus  a  rubber 
band  resists  stretching,  a  beam  resists  bending,  a  rod 
resists  twisting,  a  gas  resists  compression,  etc.  The  resist- 
ing force  developed  in  a  body,  in  consequence  of  strain,  is 
called  a  stress.  A  stress  is  a  mutual  action  between 
adjacent  parts  of  a  body.  In  a  block  of  wood  compressed 
between  the  jaws  of  a  vise  the  stress  is  a  mutual  pressure 
between  adjacent  parts  of  the  block.  In  a  stretched  wire 
the  stress  is  a  pull  or  tension,  acting  in  both  directions, 
from  end  to  end  of  the  wire.  It  tends  at  every  point  to 
pull  the  wire  apart. 

In  many  cases  the  internal  force,  or  stress,  may  be 
regarded  simply  as  the  applied  force  transmitted  through  the 
body.  Thus  the  tension  in  a  rope  is  a  transmitted  pull 
exerted  at  both  ends  of  it,  and  the  elastic  force  of  a  com- 
pressed fluid  is  a  transmitted  pressure.  In  such  cases  the 
external  or  applied  force  is  often  called  the  stress. 

88.  Elasticity  and  Elastic  Limit.  —  A  body  that  recovers 
from  a  strain  after  the  applied  forces  cease  to  act  is  said  to 
have  elasticity  or  to  be  elastic.  There  are  two  kinds  of 
elasticity,  namely,  elasticity  of  volume,  shown  by  recov- 
ery of  volume  after  compression,  and  elasticity  of  form, 
shown  by  recovery  of  shape  after  any  deformation,  whether 
of  compression,  stretching,  twisting,  or  bending.  All 
fluids  have  perfect  elasticity  of  volume;  that  is,  however 
great  the  compression,  they  always  expand  to  their  original 
volume  when  the  pressure  is  removed.  All  solids  possess 
elasticity  of  volume  in  some  degree;  but  they  may  be  per- 
manently diminished  in  volume  to  a  slight  extent  by  the 
application  of  sufficient  pressure.  The  rolling  and  stamp- 
ing to  which  silver  is  subjected  in  the  process  of  coining 


94  STATICS  OF  SOLIDS 

diminishes  its  volume  by  about  4  %.  Fluids  obviously 
have  no  elasticity  of  form;  some  solids  possess  this  prop- 
erty, others  do  not.  Putty,  wet  clay,  and  lead  are  good 
examples  of  inelastic  or  plastic  solids. 

The  elasticity  of  a  body  is  greater  or  less  in  proportion 
to  the  elastic  force  (stress)  with  which  the  body  tends  to 
recover  from  a  given  change  of  form  or  volume  (strain). 
Thus  liquids  have  greater  elasticity  than  gases,  and  the 
elasticity  of  ivory,  glass,  or  steel  is  great  compared  with 
that  of  rubber. 

An  elastic  solid  recovers  its  shape  completely  after  stretching, 
bending,  or  other  distortion,  provided  the  strain  does  not  exceed  a 
certain  limit,  which  is  known  as  the  limit  of  perfect  elasticity,  or, 
more  briefly,  the  elastic  limit  of  the  body.  A  brittle  substance,  e.g. 
glass,  breaks  when  strained  beyond  its  elastic  limit,  a  tough,  malleable, 
or  ductile  substance,  e.g.  copper  or  steel,  is  more  or  less  plastic  beyond 
its  limit  of  elasticity,  and  when  this  limit  is  exceeded  it  suffers  a 
permanent  change  of  shape.  The  elastic  limit  varies  greatly  with 
different  substances.  Among  the  metals  it  is  smallest  with  lead  and 
greatest  with  steel;  hence  steel  makes  the  best  springs,  while  lead  is 
absolutely  worthless  for  this  purpose.  Rubber  has  the  greatest 
elastic  limit  of  all  solids.  A  rubber  band  returns  to  its  original  length 
after  it  is  stretched  to  seven  or  eight  times  that  length. 

It  should  be  noted  that  the  scientific  meaning  of  the  terms  "  highly 
elastic,"  "more  elastic,"  etc.,  is  different  from  their  common  mean- 
ing. In  the  usual  sense  of  the  term,  rubber  and  air  are  highly  elastic; 
in  the  language  of  science  they  have  little  elasticity  but  a  large  elastic 
limit. 

89.  Tensile -Stress  and  Strain.  Hooke's  Law. — The  relation 
between  tensile  stress  and  strain  is  studied  experimentally  by  accu- 
rately measuring  the  elongation  of  a  long  wire  when  suspended  from 
a  fixed  support  and  stretched  by  attaching  weights  of  different  sizes 
to  its  lower  end.  The  experiment  shows  that,  so  long  as  the  strain 
is  within  the  elastic  limit  of  the  wire,  the  elongation  is  proportional 
to  the  stretching  force,  or  the  stress  is  proportional  to  the  strain.  This 
is  known  as  Hooke's  law. 


STRESSES    AND    STRAINS  95 

The  total  elongation  of  a  wire  or  rod,  under  a  given  tension,  is 
proportional  to  its  length;  for  equal  parts  of  its  length  are  equally 
stretched.  Other  conditions  remaining  the  same,  the  tension  required 
to  produce  a  given  elongation  is  proportional  to  the  cross-section  of 
the  wire  or  rod.  With  double  the  cross-section,  for  example,  the 
stretching  force  must  be  doubled,  just  as  if  two  wires  of  the  original 
size  were  placed  side  by  side  and  stretched  at  the  same  time.  Other 
conditions  remaining  the  same,  the  force  required  to  produce  a  given 
strain  varies  with  the  material.  It  is  greater  for  steel  than  for  any 
other  material;  which  is  only  another  way  of  saying  that  steel  is 
the  most  rigid  of  substances.  Engineers  assume  that  steel  can  safely 
sustain  a  tension  of  16,000  to  40,000  Ib.  to  every  square  inch  of  cross- 
section,  in  bridges  and  other  structures,  the  limit  varying  with  the 
quality  of  the  steel. 

90.  Compressive  Stress  and  Strain.  —  A  compressive  stress 
shortens  a  rod  or  beam  as  much  as  an  equal  tensile  stress  elongates 
it;  but  the  elastic  limit  of  some  materials  is  much  greater  for  com- 
pression than  it  is  for  extension.  This  is  the  case  with  cast  iron,  and 
also,  in  a  marked  degree,  with  concrete  (a  mixture  of  broken  stone, 
sand,  and  cement).  Builders  and  engineers  consider  50  Ib.  to  the 
square  inch  as  the  limit  of  safety  for  concrete  under  tension,  while 
for  compression  a  limit  of  500  Ib.  to  the  square  inch  is  allowed.  A 
maximum  of  16,000  Ib.  to  the  square  inch  is  considered  safe  for  both 
tensile  and  compressive  stresses  in  ordinary  structural  steel. 

A  compressive  stress  makes  demands  upon  the  stiffness,  rather 
than  the  strength,  of  slender  rods  and  beams.  Thus  a  slender  cane 
bends  when  lightly  pushed  against  the  ground,  although  a  short  piece 
of  it  would  transmit  a  much  greater  pressure  without  giving  way. 
In  the  transmission  of  tensions,  on  the  other  hand,  stiffness  is  not 
required,  the  utmost  flexibility,  as  in  chains  and  ropes,  being  permissi- 
ble. To  illustrate  further,  the  wooden  spokes  of  a  wagon  wheel  hold 
the  rim  and  tire  in  position  by  an  outward  thrust  or  push  from  the 
hub.  They  thus  sustain  a  compressive  stress,  and  require  stiffness, 
which  is  afforded  by  their  large  cross-section.  The  slender  steel 
spokes  of  a  bicycle  wheel,  on  the  contrary,  hold  the  rim  in  position 
by  an  inward  pull  toward  the  hub.  They  are  under  tension;  and  the 
requirement  to  be  met  is  strength,  not  stiffness. 


STATICS  OF  SOLIDS 


FlG- 


96 

91.  Strains  Produced  by  Transverse  Forces.  —  The  change  of 
length  produced  by  tensile  or  compressive  stresses  is  so  slight  that  it 
escapes  detection  under  ordinary  circumstances;  but  the  effects  of 
transverse  forces  are  often  conspicuous.  The  bending  or  sagging 
of  telephone  wires,  stretched  ropes,  horizontal  ^  —  -^ 

beams,  the  branches  of  trees,  etc.,  under  their 
own  weight,  and  the  bending  of  tree  tops  in  a 
strong  wind  are  familiar  examples.  The  pupil 
will  find  it  instructive  to  make  the  following  ex- 
perimental  study  of  the  stresses  and  strains  developed  in  bending. 
Mark  a  row  of  dots  exactly  half  an  inch  apart  on  both  sides  of  a 

long  rubber  eraser,  then  measure  the 
distance  between  the  dots  on  both 
the  inside  and  the  outside  while 
the  eraser  is  bent  with  the  fingers. 
(The  measuring  may  be  done  by 
means  of  a  strip  of  paper  on  which 
half  -inch  spaces  are  marked.)  It 
will  be  found  that  the  outer  side 
of  the  eraser  is  stretched  and  the 
inner  side  compressed  in  bending 
(Fig.  83).  The  outer  half  of  the 
eraser  is  under  a  tensile  stress, 


FIG.  84. — Bending  Beams. 


which  increases  from  zero  at  the  mid-plane  to  a  maximum  at 
the  outer  surface;  and  the  inner  half  is  under  a  compressive 
stress,  which  increases  from  zero  at  the  mid-plane  to  a  maximum 
at  the  inner  surface.  As  the  bending  is  increased,  the  tensile 
strength  of  the  rubber  will  finally  be  exceeded  at  the  outer  sur- 
face, and  it  will  tear.  The  compression  of  the  inner  half  makes 
demands  upon  the  crushing  strength  of  the  rubber,  which  is  great 
enough  to  withstand  the  strain. 

By  attaching  different  weights  to  beams,  supported  in  either  man- 
ner shown  in  Fig.  84,  it  is  found  that  the  deflection  is  proportional  to 
the  force.  (Hooke's  law.)  By  comparing  the  deflections  of  beams 
of  the  same  material,  and  of  equal  length  and  depth  but  unequal 
width,  it  is  found  that,  with  a  given  load,  the  deflection  is  inversely 
proportional  to  the  width;  or  (since  stiffness  is  inversely  proportional  to 
the  amount  of  bending) ,  the  stiffness  of  'a  beam  subjected  to  a  vertical 
load  varies  directly  as  its  width:  Thus  a  beam  5  in.  wide  (Fig.  85  b) 


STRESSES    AND    STRAINS 


97 


FIG.  85. — Horizontal  Beams,  Flat  and 
on  Edge. 


is  equivalent  to  five  beams  of  the  same  material,  length,  and  depth, 
and  i  in.  wide,  placed  side  by  side. 

By  a  similar  experiment  it  is  found  that  the  deflection  of  a  beam 
varies  inversely  as  the  cube  of  its 
depth;  or,  the  stiffness  of  a  beam 
varies  directly  as  the  cube  of  its 
depth.  Thus  a  beam  5  in.  in 
depth  (Fig.  85  c]  is  125  times  as 
stiff  as  a  beam  of  the  same 
length  and  width  and  i  in.  in 
deptH  (Fig.  85  a),  and  is  25  times 
as  stiff  in  this  position  as  it 
would  be  if  laid  broadside,  as  in  b.  The  strength  of  a  beam  does 
not  increase  as  rapidly  with  increase  of  depth  as  the  stiffness  does, 
the  ratio  being  as  the  square  of  the  depth.  Thus  while  the  beam 
c  is  125  times  as  stiff  as  the  beam  a,  it  will  sustain  only  25  times 
as  great  a  load  before  breaking,  and  will  bend  only  one  fifth  as  far  as 
a  will  bend  before  breaking. 

Other  conditions  being  the  same,  the  deflection  of  a  beam  varies 
directly  as  the  cube  of  its  length.  This  can  be  determined  experi- 
mentally by  varying  the  distance  between  the  points  of  support. 

92.  The  Best  Distribution  of  Material  in  Beams.  —  Since  the  cen- 
tral part  of  a  beam  is  strained  least  under  the  action  of  transverse 
forces,  it  contributes  least  to  the  stiffness  and  strength  of  the  beam. 
Consequently  a  horizontal  beam  or  girder,  containing  a  given  amount 
of  material,  is  made  stiffer  and  stronger  by  placing  most  of  the 
material   at   top    and   bottom,   in  the   form   of  flanges  (Fig.  86). 

Where  stiffness  is  demanded  in  all  directions,  the 
material  is  disposed  about  a  hollow  center,  in  the 
form  of  rectangular  or  circular  tubes  and  columns, 
as  in  the  frame  of  a  bicycle,  the  columns  of 
steel-frame  buildings,  and  the  "  compression  mem- 
bers" of  steel  bridges.  Nature  utilizes  this  prin- 
ciple in  the  hollow  stalks  of  grain  and  bamboo,  the 
hollow  quills  of  feathers,  etc. 

93.  The  Four  Structural  Units.  —  The  laws  of  stresses  and  strains 
determine  the  four  structural  units  employed  in  architecture  and  civil 
engineering.     These  are  the  lintel,  the  arch  or  vault,  the  truss,  and 
the  suspension  cable. 


FIG.  86.  —  Steel 
Rail. 


98 


STATICS  OF  SOLIDS 


The  simplest  and  earliest  of  these  is  the  lintel,  which  consists  of 
a  horizontal  cross-piece  or  beam,  resting  on  a  vertical  support  at 
each  end  (Figs.  87  and  88).  The 
structural  principle  of  the  lintel  is 
that  of  inert  stability.  The  mem- 
bers are  held  in  position  by  their 
own  weight  and  the  vertical  pres- 
sure of  the  load  which  they  sus- 
tain. The  lintel  was  employed 


FIG.  87.  —  Gateway  in  an  Ancient 
City  Wall  at  Mycenae,  Greece. 


FIG.  88. — The  Lintel  of  Ancient 
Grecian  Architecture. 


by  primitive  man,  and  interesting  examples  dating  from  prehistoric 
times  still  exist  (Fig.  87).  Grecian  architecture  was  based  exclu- 
sively on  the  constructive  principle  of  the  lintel.  The  huge 
blocks  of  marble,  of  which  the  temples  and  palaces  were  constructed, 
rested  one  upon  another,  requiring  no  other  element  of  stability 
than  their  own  massiveness.  The  columns  were  built  up  of  several 
round,  flat-topped  blocks,  and  the  space  between  them  was  bridged 
at  the  top  by  a  single  stone  (Fig.  88). 

94.  The  Arch.  — A  true 
arch  is  a  structure  of 
masonry  built  out  of  wedge- 
shaped  stones,which  support 
one  another  by  mutual  pres- 
sure, and  together  span  an 
opening  (Fig.  89).  An  arch 
sustains  the  downward  pres- 
FIG.  89.  —  A  True  Arch.  sure  due  to  its  own  weight 


STRESSES    AND    STRAINS 


99 


and  the  weight  of  the  overlying  structure,  and  transmits  this  pres- 
sure, partly  as  a  downward  pressure  and  partly  as  an  outward 
pressure  or  lateral  thrust  (Fig.  90).  An  arch,  therefore,  tends 


FIG.  90.  —  Distribution 
of  Stresses  in  an  Arch. 


FIG.  91. — A  Lintel  in 
the  Form  of  an  Arch. 


FIG.  92. —  An  Offset 
Arch. 


to  spread,  and,  having  no  strength  in  itself  to  resist  spreading, 
it  will  fail  unless  it  receives  adequate  support  from  the  sides  as 
well  as  from  beneath.  Where  an  arch  occurs  in  a  wall,  the  wall 
itself  provides  the  necessary  lateral  support.  Arches  forming  a  con- 
tinuous series  support  one  another,  except- 
ing, of  course,  the  outer  side  of  the  end 
ones.  An  arch  cut  out  of  a  single  stone 
(Fig.  91)  is  an  arch  only  in  shape;  struc- 
turally it  is  a  lintel,  the  stresses  between 
its  members  being  vertical  only.  The 
same  is  true  of  the  offset  arch  (Fig.  92). 

The  arch  was  first  extensively  used  by 
the  Romans,  and  was  well  adapted  to 
their  building  materials,  which  were  prin- 
cipally brick  and  stone  of  moderate  size. 
The  Roman  arch  was  always  round.  The 
pointed  arch  (Fig.  93),  which  is  a  charac- 
teristic feature  of  the  Gothic  architecture 
of  the  twelfth  and  thirteenth  centuries, 
has  the  structural  advantage  of  exerting  a 
smaller  lateral  thrust  than  the  round  arch. 
The  ceilings  of  Gothic  cathedrals  were 
built  of  thin  slabs  of  stone,  supported  by 
arches  which  bridged  the  space  to  be 
covered.  These  vaulted  ceilings  exerted  FIG.  93-  —  Flying  Buttress. 


IOO 


STATICS  OF  SOLIDS 


enormous  lateral  thrusts  against  the  walls,  which  they  alone  were 
wholly  inadequate  to  sustain;  hence  the  necessity  for  the  "flying 
buttress"  or  arched  stone  brace  to  support  the  wall  from  without, 
as  shown  in  the  figure. 

95.  The  Truss.  —  A  truss  is  a  framed  structure  which  acts  as  a 
single  rigid  body,  owing  to  the  tensile  and  compressive  stresses  sus- 
tained by  its  members.  The 
simplest  form  of  roof-truss  (Fig. 
94)  has  six  members,  namely,  two 
rafters,  AC  and  CB,  a  tie-beam, 
AB,  a  king-post,  CD,  and  two 

struts,   DF  and   DE.     The    two 
FIG.  94. — Roof -truss.  r 

rafters  are  under  a  compressive 

stress,  due  to  their  own  weight  and  the  weight  of  the  roof,  and 
tend  to  spread  at  the  bottom;  hence  they  would  exert  an  out- 
ward thrust  on  the  walls,  if  they  were  not  held  by  the  tie-beam 
AB.  The  tie-beam  is  thus  under  tension.  Its  own  weight,  how- 
ever, would  cause  it  to  sag  if  it  were  not  supported  at  the  center 
by  the  upward  pull  of  the  king-post.  The  king-post,  being  under 
tension  and  suspended  from  the  rafters,  pulls  down  on  them,  thus 
adding  to  their  compression  load.  The  rafters  would  sag  if  they  were 
supported  only  at  top  and  bottom;  but  .this  is  prevented  by  the  out- 
ward thrust  of  the  struts  at  E  and  F.  Since  the  struts  are  supported 
at  D,  they  add  to  the  tension  load  on  the  king-post;  and  this  again 
adds  to  the  compression  load  on  the  rafters,  which,  in  turn,  adds  to 
the  tension  load  on  the  beam.  This  completes  the  round  of  mutual 
actions.  (Which  are  the  tension  and  which  the  compression  mem- 
bers? Could  the  king-post  be  made  of  a  slender  iron  rod?  Could 
the  struts?) 

A  jointed  frame  having  more  than  three  sides  can  be  given  any 
number  of  shapes  by  merely  varying  the  angles;  but  a  jointed  tri- 
angular frame  is  rigid. 
Hence  a  framed  struc- 
ture built  up  of  any 
number  of  triangles  is 

rigid,  and   constitutes  a  FIG.  95.  — Pratt  Bridge  Truss, 

truss.     The   usual   form 
of  truss  for  bridges  having  a  span  of  100  to  600  ft.  is  the  Pratt  truss 


STRESSES    AND    STRAINS 


1 01 


(Figs.  95  and  96).  A  truss  supported  at  its  ends  tends  to  sag, 
like  the  lower  beam  in  Fig.  84;  hence  its  upper  horizontal  member 
is  under  a  compressive  stress,  and  its  lower  horizontal  member  is 
under  tension.  In  the  diagram  all  compression  members  (the  upper 
horizontal  member,  the  vertical  struts,  and  the  end  braces)  are  repre- 


FIG.  96.  —  The  Spans  are  Pratt  Trusses. 

sented  by  double  lines,  and  all  tension  members  (the  lower  horizontal 
member  and  the  diagonal  tie-rods)  by  single  lines.  The  tie-rods  are 
constructed  of  wrought  iron  or  steel,  and  the  compression  members 
of  rigid  timber  or  steel  beams. 

Trusses  are  made  in  the  form  of  an  arch  for  roofs  and  bridges  where 
the  span  is  long.  The  largest  arch-truss  bridge  in  the  world  crosses 
the  East  River  in  a  single  span  of  1000  feet.  An  arch-truss  exerts 


FIG.  97. — A  Cantilever  Bridge  in  Construction. 

an  outward  thrust,  which  is  sustained  by  the  abutments  or  by  a  tie- 
rod,  extending  across  between  its  lower  ends. 

A  truss  supported  at  one  end  only  is  called  a  cantilever  (Fig.  97). 
The  lower  member  of  a  cantilever  is  under  a  compressive  stress  and 
the  upper  member  is  under  tension  (like  the  upper  beam  in  Fig.  84). 
A  span  of  a  cantilever  bridge  consists  of  two  cantilevers  extending 


IO2 


STATICS  OF  SOLIDS 


out  toward  each  other  from  adjacent  piers,  but  not  meeting,  with  an 
independent  truss  suspended  between  their  ends.  The  principal 
advantage  of  this  type  of  bridge  is  that  it  can  be  built  without  the  use 
of  false-work  to  support  it  during  construction,  as  shown  in  the  figure. 
Long  spans  are  also  possible.  The  cantilever  bridge  across  the  Firth 
of  Forth  in  Scotland  has  a  span  of  1710  feet. 

96.    The  Suspension  Cable.  —  The  use  of  the  cable  in  permanent 
structures  is  limited  to  the  suspension  bridge   (Figs.  98  and  99). 

Suspension  bridges  of  crude  form  date 
from  the  earliest  times,  and  the  first 
long  span  bridges  were  of  this  type. 
Since  the  cable  sustains  the  weight 
of  the  span  and  the  load  upon  it,  it 
is  subjected  to  an  enormous  tension 
and  must  be  firmly  anchored  at  each 
end.  A  truss  is  added  to  give  the 
necessary  stiffness  to  the  bridge  and 
to  transmit  the  load  uniformly  to  the 
cable. 

The  Manhattan  suspension  bridge 
across  East  River  is  the  heaviest 
bridge  in  existence.  It  carries  eight 


FIG.  98.  —  A  Primitive  Suspension 
Bridge  of  Bamboo  in  Hindustan. 


railroad  tracks  in  addition  to  a  wide  roadway  for  vehicles  and  two 
foot-paths   for  pedestrians.     The  main  span  is  1470  ft.  long,   and 


FIG.  99.  —  Niagara  Falls  Railway  Suspension  Bridge. 

the  two  side  spans  725  ft.  each.  The  total  pull  of  the  four  cables 
amounts  to  30,000  tons,  and  their  anchorage  at  each  end  of  the 
bridge  contains  233,000  tons  of  stone  and  concrete.  The  cables 


STRESSES    AND    STRAINS  103 

are  21  in.  in  diameter,  and  each  of  the  four  contains  9492  wires,  yV 
of  an  inch  in  diameter.  The  total  length  of  single  wire  in  all  the 
four  cables  is  23,100  mi.,  or  nearly  sufficient  to  girdle  the  earth. 

PROBLEMS 

1.  A  2 -in.  by  1 2 -in.  joist  is  how  many  times  as  stiff  and  how  many  times 
as  strong  in  the  correct  position  as  it  is  when  placed  on  its  broad  side? 

2.  How  does  a  beam  2  in.  wide  and  6  in.  deep  compare  in  stiffness  and 
strength  with  a  beam  of  the  same  material,  length,  and  cross-section,  but 
3  in.  wide  and  4  in.  deep? 

3.  In  concrete  buildings  the  tensile  stresses  are  borne  by  steel  rods, 
imbedded  in  the  concrete.     Where  must  these  rods  be  placed  in  horizontal 
beams  to  serve  this  purpose  at  all?     Where  must  they  be  placed  to  be  of 
the  greatest  service? 


CHAPTER  VI 

DYNAMICS 

97.  Dynamics  may  be  briefly  defined  as  the  mechanics 
of  unbalanced  forces.     Stated  more  fully,  it  is  that  branch 
of  mechanics  which  treats  of  the  relations  between  unbal- 
anced force,  the  mass  acted  upon,  and  the  motion  or  change 
of  motion  produced.     It  will  simplify  matters  to  begin 
with  a  purely  mathematical  study  of  some  important  types 
of  motion,  considered  independently  of  the  mass  of  the 
body  moved  or  the  cause  of  its  motion. 

I.   MOTION 

98.  Motion  is  Change  of  Position. — A  body  is  in  motion 
relatively  to  another  body  if  either  its  distance  or  its  direc- 
tion from  that  body  is  changing.     If  both  its  distance  and 
its  direction  from  the  other  body  remain  unchanged,  the 
bodies  are   at  rest  relatively  to  each  other.     A   person 
seated  in  a  moving  car  is  at  rest  with  reference  to  the 
car,  but  in  motion  with  reference  to  the  earth.     A  person 
standing  on  the  ground  is  at  rest  with  respect  to  the  earth; 
but  with  respect  to  the  sun  and  the  planets  he  is  in  rapid 
motion  with  the  earth  in  its  dcfily  rotation  and  its_annual 
revolution.     It  is  evident  that  rest  and  motion  are  relative 
terms.     Unless  otherwise  stated  or  clearly  implied,  it  is 
understood  that  rest  and  motion  are  expressed  relatively 
to  the  earth. 

The  line  along  which  the  center  of  gravity  of  a  body 
moves  is  regarded  as  the  path  of  the  body.     The  motion 

104 


MOTION  105 

of  a  body  is  completely  known  when  its  path  and  its  rate 
of  motion  at  every  point  of  the  path  are  known,  or  the  rate 
and  direction  of  motion  at  every  instant  during  the  motion. 
Rate  of  motion  is  called  speed.  Velocity  includes  both 
speed  and  direction  of  motion.  Thus  "  six  miles  per  hour  " 
is  the  complete  description  of  a  speed;  "six  miles  per 
hour  toward  the  east  "is  the  complete  description  of  a 
velocity.  The  distinction  between  the  two  words  is  fre- 
quently useful,  but  is  not  strictly  adhered  to,  the  term 
velocity  being  frequently  used  where  speed  alone  is  under 
consideration. 

99.  Uniform  Motion.  —  If  the  speed  of  a  body  is  con- 
stant, the  motion  is  said  to  be  uniform,  and  is  measured 
by  the  distance  that  the  body  moves  over  in  a  unit  of 
time.  The  principal  metric  unit  of  speed  is  i  cm.  per 
second,  and  the  principal  English  unit  is  i  ft.  per  second. 
Speed  is  often  expressed  in  meters  per  second,  miles  per 
hour,  miles  per  minute,  or  even  in  miles  per  second  when 
very  great. 

The  whole  distance  passed  over  by  a  body  moving  with 
constant  speed  is  equal  to  the  product  of  the  gpppd  a^d  th** 
time  occupied  in  traversing  the  distance.  Hence,  letting  d 
denote  the  distance,  v  the  magnitude  of  the  velocity,  and 
t  the  time,  we  have  for  uniform  motion —  B> 

d  =  vtj  also  v  =-,  and  t  =  -.  (i) 

o{ — >-<« 

100.  Representation  and  Composition  of  FlG- 10° 
Velocities.  —  A  velocity  may  be  represented  both  in  mag- 
nitude and  direction  by  a  straight  line,  just  as  a  force  may 
be.  Thus  if  OB  (Fig.  100)  represents  a  velocity  of  2  ft. 
per  second  toward  the  north,  then  OA  represents  a 
velocity  of  3  ft.  per  second  toward  the  east. 


106  DYNAMICS 

A  body  may  have  two  or  more  independent  motions  at 
the  same  time.  Thus  a  boat  rowed  across  a  stream  has 
a  motion  imparted  by  the  rowing,  and  also  a  motion  due 
to  the  current  and  equal  to  it. 

Suppose  the  boat  to  be  constantly  headed  directly 
toward  the  opposite  shore,  and  let  O  (Fig.  101)  represent 
the  starting  point.  OB  would  be  the  path  of  the  boat  if 
there  were  no  current.  OC  is  the  dis- 
tance the  stream  flows  while  the  boat  is 
crossing.  The  actual  motion  of  the  boat 
relative  to  the  earth  is  the  resultant  of 
these  two  motions,  and  its  path  is  repre- 
sented by  OA.  If  OB  and  OC  be  taken 
to  represent  the  component  velocities  (in- 
FIG.  101.  stead  Of  tne  wnoie  distances),  then  OA, 
which  is  the  concurrent  diagonal  of  the  parallelogram 
constructed  on  OB  and  OC  as  sides,  will  represent  the 
actual  or  resultant  velocity  upon  the  same  scale.  Veloc- 
ities are,  in  fact,  compounded  by  the  same  rules  as 
forces,  and  the  construction  is  called  the  parallelogram 
of  velocities. 

101.  Resolution  of  a  Velocity.  —  A  velocity,  like  a  force, 
can  be  resolved  into  components  in  any  chosen  directions, 
and  the  construction  is  the  same  as  for  the  resolution  of  a 
force.  For  example,  suppose  we  wish  to 
know  at  what  rate  a  vessel  is  advancing 
northward  and  at  what  rate  eastward,  when 
it  is  sailing  30°  north  of  east  at  a  rate  of  FIG.  102. 
12  mi.  per  hour. 

If  OA  (Fig.  102)  represents  the  velocity  of  the  vessel, 
ON  and  OE  will  represent  its  northerly  and  easterly  com- 
ponents, respectively,  to  the  same  scale.  Now  it  is  proved 


MOTION  107 

in  geometry  that  in  a  right  triangle,  having  an  acute  angle 
of  30°,  the  hypothenuse  is  twice  the  shorter  side.  Hence 
ON  represents  a  velocity  of  6  mi.  per  hour,  and  OE  a 
velocity  of  V  i22  —  62,  or  10.4  mi.  per  hour. 

As  a  further  illustration,  let  us  consider  how  the  boat  mentioned 
in  the  preceding  article  must  be  rowed  in  order  to  reach  the  opposite 
bank  at  B  instead  of  A .  '  The  resultant  motion  is  now 
represented  by  OB.  The  component  OC,  due  to  the 
motion  of  the  stream,  is  the  same  as  before.  Hence 
OB  (Fig.  103)  is  the  diagonal  of  a  parallelogram  of 
which  one  side  is  OC.  The  other  component  motion 
is  therefore  represented  by  OA'.  This  means  that 
the  boat  must  be  constantly  pointed  in  a  direction 
parallel  to  OA ',  and  that  it  would  take  as  long  to 
reach  B  as  it  would  to  row  the  distance  OA '  in  still 
water. 

PROBLEMS 

1.  A  velocity  of  50  mi.  per  hour  is  a  velocity  of  how  many  feet  per 
second? 

2.  A  train  runs  with  a  velocity  of  23  m.  per  second.     In  what  time 
does  it  run  a  kilometer? 

3.  From  a  train  running  at  the  rate  of  9  m.  per  second,  a  mail-bag  is 
thrown  at  right  angles  to  the  track  with  a  velocity  of  4  m.  per  second.    Com- 
pute the  resultant  velocity  of  the  bag  at  the  instant  it  leaves  the  hand, 
and  draw  a  figure  to  show  its  direction. 

4.  From  a  train  running  at  the  rate  of  12  m.  per  second,  a  mail-bag  is 
thrown  so  that  its  resultant  velocity  is  equal  to  that  of  the  train  and  at  right 
angles  to   it.     What  is  the   magnitude   and  direction  of  the  velocity  im- 
parted in  throwing  the  bag? 

6.  An  arrow  is  shot  directly  backward  from  the  rear  of  a  train  with  a 
velocity  (relative  to  the  train)  equal  to  that  of  the  train.  What  is  the 
motion  of  the  arrow? 

6.  The  rotation  of  the  earth  carries  its  surface  eastward  at  the  rate  of 
about  \  mi.  per  second  (in  temperate  latitudes).  When  a  ball  is  thrown 
up,  why  is  it  not  left  behind  (to  the  west)  by  the  earth  in  its  rotation? 


io8  *  DYNAMICS 


7.  Four  boys,  A,  B,  C,  and  D  (Fig.  104),  on  the  deck  of 
a  moving  vessel,  pass  a  ball  round  in  the  order  of  the  letters. 
What  allowance  for  the  motion  of  the  vessel,  if  any,  must  be 
made  by  each  of  the  boys  in  throwing?     Give  reasons. 

8.  A  vessel  sails  due  N.  E.  at  the  rate  of  15  mi.  per  hour. 
FIG.  104.        Compute    the    northerly    and    easterly  components  of   its 

velocity. 

9.  A  boatman  wishes  to  cross  a  stream  where  it  is  100  m.  wide 
and  its  velocity  .8  m.  per  second.  The  component  velocity  that  he  imparts 
to  the  boat  by  rowing  is  1.2  m.  per  second,  (a)  How  long  will  it  take  him 
to  cross  the  stream  to  a  point  directly  opposite  to  the  starting  point  (Fig. 
103)?  (b)  How  long  will  it  take  him  to  cross  if  he  rows  as  shown  in  Fig.  101? 

10.  A  boy  is  riding  north  with  a  velocity  of  12  mi.  per  hour,  (a)  What  is 
the  apparent  direction  and  velocity  of  the  wind  if  the  air  is  still?  (b)  What 
if  there  is  an  east  wind  of  20  mi.  per  hour? 

102.  Variable  Motion.  —  If  the  motion  of  a  body  is 
variable,  qualifying  terms  must  be  used  in  specifying  its 
velocity,  such  as  its  velocity  at  a  certain  instant,  or  at  a  cer- 
tain point  of  its  path,  or  its  average  velocity  during  a  speci- 
fied interval  of  time,  etc.  The  unqualified  term  "velocity" 
refers  to  no  one  of  these  velocities  in  particular,  and  hence 
has  no  definite  or  intelligible  meaning.  For  example,  the 
language  is  definite  when  we  say  that  the  velocity  of  a  fall- 
ing body  is  128.6  ft.  per  second  at  the  end  of  the  fourth 
second  from  the  start. 

The  velocity  of  a  body  at  a  given  instant  is  measured  by 
the  distance  it  would  pass  over  during  the  following  unit 
of  time,  if  its  velocity  continued  unchanged  from  that 
instant.  Thus  when  we  say  that  a  train  is  running  at  the 
rate  of  30  mi.  per  hour,  we  mean  that  it  would  run  30 
miles  in  an  hour  if  it  continued  at  its  existing  rate  for  one 
hour.  The  average  velocity  of  a  body  during  any  inter- 
val of  time  is  equal  to  the  uniform  velocity  required  to 
cover  the  same  distance  in  the  same  time.  Thus  if  an 


MOTION  109 

automobile  goes  108  mi.  in  6  hr.,  its  average  rate  is  18  mi. 
per  hour,  since  this  is  the  uniform  rate  required  to  run  the 
given  distance  in  the  given  time.  The  actual  velocity  may 
vary  from  zero  (during  intervals  of  stopping)  to  40  or  more 
miles  per  hour.  It  follows  from  the  definition  that  aver- 
age velocity  is  equal  to  the  distance  divided  by  the  time. 
Representing  average  velocity  by  v,  its  definition  is 
expressed  by  the  formula  — 

d  =  vt ;  from  which  v  =  —  (2) 

•» 

103.  Acceleration.  —  A  change  of  velocity  may  consist 
in  a  change  (increase  or  decrease)  of  speed,  or  a  change  in 
the  direction  of  motion,  or  in  a  change  both  of  speed  and 
direction.  The  term  acceleration,  in  its  general  sense, 
includes  all  of  these  possible  changes  in  velocity.  For 
motion  in  a  straight  line,  acceleration  consists  in  a  change 
of  speed  only,  and  is  uniform  or  constant  if  the  speed  in- 
creases or  decreases  uniformly  with  the  time  (not  with  the 
distance  passed  over).  Uniform  acceleration  in  the  line 
of  motion  is  measured  by  the  change  of  speed  per  second. 
For  example,  the  speed  of  a  body  is  uniformly  accelerated 
if  it  increases  by  3  m.  per  second  every  second.  At  the  end 
of  the  first  second  the  speed  would  be  3  m.  per  second;  at 
the  end  of  2  sec.  it  would  be  6  m.  per  second;  at  the  end 
of  3  sec.,  9  m.  per  second;  etc.  The  motion  in  this  case 
is  said  to  be  accelerated  at  the  rate  of  3  m.  per  second 
per  second.  The  repetition  of  the  phrase  "per  second"  is 
necessary,  for  time  is  doubly  involved  in  acceleration. 
With  an  acceleration  of  3  m.  per  second  per  second,  a 
velocity  of  180  m.  per  second  is  gained  in  a  minute;  hence 
this  might  be  expressed  as  an  acceleration  of  180  m.  per 
second  per  minute. 


no  DYNAMICS 

The  principal  metric  unit  of  acceleration  is  an  accelera- 
tion of  i  cm.  per  second  per  second;  the  principal  English 
unit  is  an  acceleration  of  i  ft.  per  second  per  second. 

Generally  the  acceleration  of  bodies  is  not  uniform  but 
variable.  A  train  gains  speed  less  and  less  rapidly  for  some 
distance  after  starting,  until  finally  the  acceleration  be- 
comes zero  and  the  speed  constant.  The  acceleration  of  a 
street  car  is  irregular,  increasing  abruptly  whenever  more 
power  is  turned  on,  and  decreasing  steadily  during  the 
intervals  between.  This  occurs  several  times  in  getting  up 
speed.  The  best  example  of  uniformly  accelerated  motion 
is  a  falling  body,  provided  its  motion  is  not  sensibly  affected 
by  the  resistance  of  the  air.  The  acceleration  of  a  falling 
body  varies  slightly  in  different  latitudes.  In  the  temper- 
ate zones  it  is  close  to  980  cm.  or  32.16  ft.  per  second  per 
second. 

It  is  important  to  understand  that  in  uniformly  accel- 
erated motion  the  acceleration  is  the  same,  not  only  for 
each  second,  but  for  any  number  of  seconds  or  any  frac- 
tion of  a  second.  In  other  words,  the  rate  of  change  of 
velocity  is  the  same  during  the  first  hundredth  part  of  a 
second  as  it  is  during  any  part  of  the  time  or  the  whole  of 
it.  Acceleration  is  called  positive  when  it  consists  in 
increase  of  speed,  negative  when  it  consists  in  decrease  of 
speed.  It  is  customary  in  common  speech  and  in  elemen- 
tary physics  to  use  the  terms  retarded  motion  and  retarda- 
tion instead  of  negatively  accelerated  motion  and  negative 
acceleration. 

104.  Digression  on  the  Average  Value  of  a  Uniformly  Changing 
Variable.  —  If  a  board  is  6  in.  wide  at  one  end  and  1 2  in.  wide  at  the 

other,  it  does  not  follow  that  its  average  width  is  —       —  or  9  in. 
By  definition,  its  average  width  is  the  uniform  width  of  a  board  of 


MOTION  in 

the  same  length  and  of  equal  area.  If  the  shape  of  the  board  is  as 
shown  in  Fig.  105,  its  average  width  is  9  in.;  for  the  piece  def  would 
fit  in  the  position  aeg,  giving  a  uniform  width  of  9  in.  We  see  also 
that  the  average  width  of  the  board  is  its  actual  width  midway  be- 


FIG.  105.  FIG.  106. 

tween  its  ends.  The  width  of  this  board  increases  uniformly  from 
one  end  to  the  other;  i.e.  starting  at  the  narrow  end,  there  is  the  same 
increase  of  width  for  each  foot  of  length.  None  of  the  above  relations 
hold  in  the  case  of  a  board  having  the  shape  abcde  (Fig.  106)  or  the 
shape  abcde'.  In  the  one  case  the  average  width  is  obviously  less 
than  the  half  sum  of  the  widths  at  the  ends,  and  in  the  other  case  it 
is  greater,  in  neither  case  is  it  equal  to  the  actual  width  midway 
between  the  ends. 

Similarly,  the  average  speed  of  a  body,  during  any  time  interval 
within  which  the  speed  changes  uniformly  with  the  time,  is  equal  to 
the  half  sum  of  the  speed  at  the  beginning  and  at  the  end  of  that  time 
interval,  and  is  also  the  actual  speed  at  the  end  of  half  that  interval. 
To  illustrate:  Since  a  falling  body  acquires  a  velocity  of  32.16  ft.  per 
second  during  each  second  of  its  fall,  its  velocity  at  the  end  of  four 
seconds  from  the  start  is  4  X  32.16  ft.,  or  128.64  ft.  per  second.  Since 
the  increase  of  velocity  is  uniform  and  the  velocity  at  the  start  is 

(0  +  128.64) 
zero,  the  average  velocity  during  the  four  seconds  is  - 

or  64.32  ft.  per  second.  This  is  also  the  actual  velocity  at  the  end  of 
half  that  interval,  or  at  the  end  of  two  seconds.  The  whole  distance 
covered  in  the  fall  is,  of  course,  the  product  of  the  average  velocity 
and  the  time,  which  is  4  X  64.32,  or  257.28  ft. 

105.   An  Experiment  on  Uniformly  Accelerated  Motion. 

-  The  motion  of  a  freely  falling  body  is  too  rapid  for  con- 
venient experimental  study.  This  difficulty  is  overcome 
by  taking  instead  the  motion  of  a  sphere  on  an  inclined 
plane.  The  acceleration  may  be  made  as  small  as  we 


112 


DYNAMICS 


please  by  diminishing  the  inclination  of  the  plane;  and,  if 
the  sphere  and  the  plane  are  as  nearly  perfect  as  may  be, 
the  acceleration  is  uniform,  as  in  the  case  of  a  freely  falling 


FIG.  1070. 


FIG.  io7&. 

body.  This  method  was  adopted  by  the  great  Italian  math- 
ematician and  scientist,  Galileo,  who  by  means  of  it  dis- 
covered the  laws  of  uniformly  accelerated  motion  in  the 
early  part  of  the  seventeenth  century. 

Figures  107 a  and  107  b  represent  convenient  devices  for 
determining  the  distances  passed  over  by  the  sphere  in  equal 
intervals  of  time.  The  first  consists  of  a  polished  inclined 
groove,  the  cross-section  of  which  is  an  arc  of  a  circle.  A 
large  steel  ball,  started  at  one  side  of  the  groove  near  the 
top,  traces  a  visible  wavy  path  in  lycopodium  powder  or 
sulphur,  dusted  over  the  groove.  This  path  is  the  result 
of  two  motions,  namely,  a  rocking  motion  from  side  to  side, 
which  serves  to  mark  equal  intervals  of  time,  and  the  accel- 
erated downward  motion.  The  motion  of  the  ball  is  so 
controlled  at  the  start  that  its  downward  motion  begins 
at  the  instant  when  it  first  crosses  the  middle  of  the  groove. 
Hence  the  distances  between  successive  points  where  the 
path  crosses  the  middle  line  of  the  groove  are  covered  in 
equal  times.  These  distances  will  be  found  to  be  in  the 


MOTION  113 

ratio  of  the  numbers  i,  3,  5,  7,  9,  n,  etc.,  within  a  small 
limit  of  experimental  error. 

The  second  form  of  apparatus  provides  four  parallel 
grooves  for  four  balls,  which  are  released  from  the  same 
height  at  the  same  instant.  When  the  stops  are  adjusted 
so  that  the  balls  are  stopped  in  succession  at  distances 
in  the  ratio  of  i,  3,  5,  and  7,  the  balls  are  heard  to  strike 
at  equal  intervals  of  time. 

106.  Analysis  of  the  Results.  —  Distances  Passed  Over  in 
Successive  Seconds.  —  The  distances  passed  over  in  equal 
times  in  either  of  the  above  experiments  are  in  the  same 
ratio  whether  the  time  interval  is  one  second  or  not.  For 
simplicity,  we  shall  assume  the  interval  to  be  one  second. 
If  the  first  distance  is  denoted  by  di  the  distances  passed 
over  in  successive  seconds  are  di  cm.,  $di  cm.,  5^  cm., 
7</i  cm.,  etc. 

Average  Velocities.  —  The  distance  passed  over  in  any 
second  is  2^1  cm.  greater  than  the  distance  passed  over  in 
the  preceding  second  (3^1  —  d\  =  2d^  $di  —  $d\  =  2di, 
'jdi  —  5^1  =  -2dij  etc.).  But,  by  definition,  the  distance 
passed  over  in  any  second  measures  the  average  velocity 
for  that  second;  hence  the  average  velocity  for  successive 
seconds  increases  at  the  rate  of  2d\  cm.  per  second. 

Acceleration.  —  The  actual  velocity  increases  at  the  same 
rate  as  the  average  velocity  for  successive  seconds;  hence 
the  acceleration  is  2di  cm.  per  second  per  second.  The 
acceleration  is  thus  numerically  equal  to  twice  the  distance 
passed  over  during  the  first  second. 

Velocity  at  the  End  of  Any  Second.  —  Since  the  velocity 
increases  at  the  rate  of  2di  cm.  per  second  every  second, 
at  the  end  of  i  sec.  the  velocity  is  2di  cm.  per  second  (or 


H4  DYNAMICS 

twice  the  average  velocity  for  the  first  second);  at  the 
end  of  2  sec.  it  is  4^1  cm.  per  second;  at  the  end  of  3  sec.  it 
is  6di  cm.  per  second;  and  at  the  end  of  /  sec.  it  is  i  X  2^1 
cm.  per  second. 

The  velocity  at  any  instant  and  the  average  velocity 
during  any  second  must  be  clearly  distinguished.  To  illus- 
trate: The  velocity  at  the  beginning  of  the  third  second  is 
4^1  cm.  per  second,  and  at  the  end  of  it  6di  cm.  per  second. 
Since  the  increase  is  uniform,  the  average  velocity  during 


! 
the   second  is  -  -  =  5^1   cm.  per  second;    hence 

5^1  cm.  is  the  distance  passed  over  during  this  second.  In 
general,  the  average  velocity  during  any  second  is  greater 
by  di  than  the  velocity  at  the  beginning  of  the  second,  and 
less  by  di  than  the  velocity  at  the  end  of  it. 

The  velocity  at  the  end  of  any  second  can  be  determined 
experimentally  by  means  of  two  planes  AB  and  BC  (Fig. 
108),  the  second  plane  being  inclined  just  enough  to  over- 
come friction,  so  that  the  speed  of  the  ball  on  it  will  be 


FIG.  108. 

constant.  The  ball  is  started  on  the  plane  AB  at  such  a 
height  that  it  reaches  the  foot  of  this  plane  in  one,  two,  or 
three  seconds,  as  desired;  and  the  distance  passed  over  in 
one  second  on  BC  measures  the  velocity  that  the  ball 
acquired  on  AB.  Thus  if  the  ball  goes  from  D  to  B  in 
one  second,  it  will  go  twice  that  distance,  or  BE,  during 
the  second  second.  (Why?) 

The  whole  distance  passed  over  from  the  start  is  d\  cm. 
in  i  sec.,  4^/1  cm.  in  2  sec.,  9^1  cm.  in  3  sec.,  i6di  cm.  in 


MOTION  115 

4  sec.,  etc.  These  distances  are  in  the  ratio  of  the  numbers 
i,  4,  9,  1 6,  etc.,  and  the  squares  of  the  time  intervals  (i2, 
22,  32,  42,  etc.)  are  in  the  same  ratio;  that  is  — 

The  whole  distance  passed  over  in  uniformly  accelerated 
motion  is  proportional  to  the  square  of  the  time. 

107.   Formulas   for   Uniformly  Accelerated    Motion.  — 

The  relations  established  above  for  uniformly  accelerated 

motion  in  a  straight  line  are  also  derived  algebraically  .as 

follows:  Let  a  denote  the  acceleration,  t  the  number  of 

seconds  during  which   the   acceleration  continues,   v  the 

velocity  at  the  end  of  that  time  (called  the  final  velocity),  v 

the  average  velocity  during  the  whole  time,  and  d  the  whole 

distance  passed  over. 

Then  v  =  at,  (Why?)  (3) 

and  v  =  \  at,          (Why?)  (4) 

and  d  =  vt  =  \  at  X  /  =  \a?\ 

that  is,  d  =  |  at2]  and  /  =    t/2^  IS) 

' 


7) 

From  equation  (3),  t  =  ~-      Substituting  this  value  of  /  in  equa- 
tion (5),  we  have 


that  is,  d  =  — >  and  v  =  ^2ad.  (6) 

2a 

These  formulas  are  algebraic  statements  of  the  laws  of  uniformly 
accelerated  motion  in  a  straight  line.  (State  these  laws  in  words, 
naming  in  full  the  quantities  for  which  the  letters  stand,  instead  of 
the  letters.)  If  any  two  of  the  three  quantities  in  any  one  of  the 
above  formulas  are  given,  the  value  of  the  third  quantity  can  be 
found  by  substituting  the  given  values  in  the  formula. 

108.  Falling  Bodies.  —  The  formulas  and  laws  for  uni- 
formly accelerated  speed  hold  for  bodies  falling  freely 


n6  DYNAMICS 

from  rest,  their  motion  being  uniformly  accelerated  in  a 
straight  (vertical)  line;  but  in  this  case  the  acceleration  is 
denoted  by  g,  instead  of  a,  since  it  is  due  to  gravity.  (Write, 
the  formulas  for  falling  bodies,  and  state  their  meaning 
in  words,  calling  g  "  the  acceleration  due  to  gravity.") 

In  numerical  work  the  value  of  g  is  to  be  taken  as  980 
cm.,  or  32.16  ft.  per  second  per  second. 

109.  Composition  of  a  Constant  and  a  Uniformly  Accelerated 
Velocity.  —  If  a  body  is  already  in  motion  when  it  begins  to  acquire 
a  uniformly  accelerated  motion,  its  velocity  at  that  instant  is  called 
the  initial  velocity,  and  its  velocity  at  any  later  instant  is  the  resultant 
of  the  initial  and  the  accelerated  velocities.  Three  cases  arise,  as 
follows: 

First  Case:  When  the  initial  and  the  accelerated  velocities  have  the 
same  direction.  — Example:  If  a  ball  is  thrown  vertically  downward, 
its  initial  velocity  as  a  freely  falling  body  is  the  velocity  imparted  in 
throwing;  i.e.  it  has  this  velocity  when  the  accelerated  motion  due 
to  gravity  begins.  Since  the  initial  velocity  and  the  velocity  acquired 
in  falling  have  the  same  direction,  their  resultant  is  their  sum. 

Let  VQ  denote  the  initial  velocity  of  a  body  which  moves  in  a 
straight  line  with  a  constant  acceleration  a,  v  its  velocity  at  the  end 
of  /  seconds,  and  d  the  distance  passed  over  in  that  time;  then 

V   =  VQ  +  at.  (7) 

Since  the  velocity  increases  uniformly  from  the  initial  velocity 
VQ  to  the  final  velocity  VQ  +  at,  the  average  velocity  v  during  the  / 
seconds  is  half  their  sum,  or 

v  =  $  [  VQ  +  (»o  +  at)  ]  =  VQ  +  \  at. 
Hence  d  =  vt  =  v0t  +  *  a/2.  (8) 

The  part  vQt  is  the  distance  the  body  would  go  in  t  seconds  with 
the  constant  initial  velocity  VQ,  and  the  part  i  at2  is  the  additional 
distance  due  to  the  accelerated  velocity. 

Second  Case:  When  the  initial  and  the  accelerated  velocities  have  op- 
posite directions.  —  Example:  A  body  thrown  vertically  upward  loses 
velocity  at  the  rate  of  980  cm.  per  second  per  second,  the  acceleration 
due  to  gravity  being  negative,  or  opposite  to  the  direction  of  motion. 


MOTION  117 

Thus  if  the  initial  velocity  were  49  m.  per  second,  the  body  would  con- 
tinue to  rise  for  5  sec.  (49  -~  9.8  =  5),  and  at  the  end  of  that  time  it 
would  come  to  rest.  Since  the  velocity  decreases  uniformly  to  zero, 
the  average  velocity  is  half  the  initial  velocity.  The  height  to  which 
the  body  rises  is  equal  to  the  product  of  the  average  velocity  and  the 
time  of  rise.  0 

A  ball  rolling  on  a  smooth,  level  surface  is  also  an  example,  its 
speed  being  uniformly  retarded  by  friction. 

The  formulas  are  as  follows,  the  letters  having  the  same  meaning 
as  above: 

v  =  VQ  —  at',  (9) 

d  =  vot  -  i  at2.  (10) 

If  /  is  the  whole  time  to  the  instant  when  the  body  comes  to  rest, 
the  final  velocity  v  is  zero,  and 

/=  -    (Why?),  and^o  =  at.  (n) 

Hence  v  =  i  VQ  =  \  at, 

and  d  =  vt  =  i  at2.  (12) 

(State  the  meaning  of  these  formulas,  and  compare  with  formulas 
3,  4,  and  5.) 

Third  Case:  When  the  initial  and  the  accelerated  velocities  are  at 
any  angle  with  each  other.  —  Example:  The  muzzle  velocity  of  a 
bullet  is  the  initial  velocity  of  its  free  flight.  The  initial  velocity 
may  have  any  direction;  the  accelerated  velocity  due  to  gravity  is 
always  vertical.  Fig.  109  shows  a  simple  device  for  an  experimental 
illustration  of  this  case.  A  slender  board  carries  a  shelf  at  one  end, 
the  other  end  is  clamped  in  a  vise.  When  the  free  end  is  drawn 
aside  and  released,  a  small  object  on  the  forward  side  of  the  shelf  is 
driven  before  it,  while,  on  the  other  side,  the  shelf  slips  from  under  a 
second  object,  which  is  thus  released  at  the  same  instant  as  the  first 
but  without  initial  velocity. 
The  two  bodies  reach  the 
floor  at  the  same  time, 
showing  that  the  acceler- 
ated motion  in  the  vertical  FlG-  IOQ- 
direction  is  the  same  for 
both.  In  whatever  direction  a  body  may  be  thrown  or  projected  into 
space,  the  accelerated  vertical  component  of  Us  motion  during  its  flight 


n8 


DYNAMICS 


is  the  same  as  that  of  a  body  falling  freely  from  rest,  the  other  component 
of  its  motion  being  the  initial  velocity. 

The  path  of  such  a  body  is  represented  graphically  in  Figs,  no 
and  in.  The  direction  of  projection,  OD,  is  horizontal  in  the  first 
case,  and  obliquely  upward  in  the  second.  In  both  figures  OA  repre- 
sents the  initial  velocity,  and  Oa  rep- 
resents i  g  on  the  same  scale.  With 
the  initial  velocity  only,  the  path  of 
the  body  would  be  represented  by  OD, 
A  B  c  D 


d' 


FIG.  no.  —  Path  of  Projectile,  Initial 
Velocity  Horizontal. 


FIG.  in.  —  Path  of  Projectile, 
Initial  Velocity  Oblique. 


and  its  position  at  the  end  of  successive  seconds  by  A,  B,  C,  and 
D.  With  the  accelerated  velocity  only,  the  path  and  the  distances 
passed  over  in  successive  seconds  would  be  as  represented  by 
Oabcd.  Hence  the  points  a',  bf,  c',  and  d'  represent  the  position 
of  the  body  at  the  end  of  successive  seconds  when  these  two  motions 
occur  at  the  same  time.  A  smooth  curve  drawn  through  these 
points  represents  the  actual  path.  The  velocity  of  the  body  at  any 
point  of  its  path,  as  at  b '  in  the  figures,  is  the  resultant  of  the  initial 
and  the  vertical  velocities,  the  latter  being  the  same  as  that  of  a  body 
starting  from  rest  and  falling  vertically  for  the  same  time.  The 
direction  of  the  resultant  velocity  is,  of  course,  tangent  to  the  curved 
path  at  every  point. 

It  should  be  noted  that  when  the  acceleration  and  the  motion  of 
the  body  are  not  in  the  same  line,  one  consequence  of  the  acceleration 


MOTION  119 

is  always  a  chdnge  of  direction  of  motion,  and  the  path  of  the  body 
is  a  curve.  There  may  or  may  not  be  a  change  of  speed  at  the  same 
time.  In  the  case  just  considered,  the  acceleration  due  to  gravity 
results  in  a  change  of  both  speed  and  direction. 

PROBLEMS  , 

1.  A  street  car  runs  with  a  constant  acceleration  of  1.2  m.  per  second 
per  second  for  8  sec.  after  starting,    (a)  What  is  its  velocity  at  the  end  of  that 
time?      (b)  What  was  its  average  velocity  during  the  8  sec.?     (c)  How  far 
does  it  run  in  the  8  sec.? 

2.  A  stone  falls  with  a  constant  acceleration  of  980  cm.  per  second  per 
second.     In  what  time  will  it  acquire  a  velocity  of  35  m.  per  second? 

3.  A  body  moves  with  a  constant  acceleration  a.      (a)  How  far  does 
it  go  in  the  first  second?     (b)  What  is  its  average  velocity  during  the  first 
second?      (c)  What  is   the  average   velocity  during  the  first  6  sec.?     (d) 
What  is  the  average  velocity  during  the  sixth  second? 

4.  A  train,  running  with  constant  acceleration,  goes  560  m.  during  the 
first  minute  after  starting.     Find  the  acceleration  in   meters  per  second 
per  second. 

6.  A  car  runs  with  a  constant  acceleration  of  80  cm.  per  second  per 
second  for  a  distance  of  300  m.  (a)  What  is,  then,  its  velocity?  (b)  With 
what  average  velocity  did  it  run  that  distance?  (c)  How  long  did  it  take  to 
run  this  distance? 

6.  A  ball  rolling  along  the  ground  is  uniformly  retarded  at  the  rate  of 
4  m.  per  second  per  second.    Its  velocity  at  the  start  is  20  m.  per  sec.    (a) 
How  long  will  it  roll?     (b)  How  far  will  it  roll? 

7.  How  far  does  a  body  fall  during  the  first. second?    Account  for  the 
fact  that  this  distance  is  numerically  equal  to  half  the  acceleration. 

8.  A  stone  dropped  from  a  cliff  strikes  the  foot  of  it  in  3.5  sec.     What 
is  the  height  of  the  cliff? 

9.  Two  stones  are  thrown  to  the  same  height,  one  vertically,  the  other 
obliquely.     Is  the  time  of  flight  the  same  for  both?     Explain. 

10.  A  stone  thrown  to  the  height  of  a  tree  reaches  the  ground  in  5  sec. 
from  the  time  of  starting.     How  high  is  the  tree? 

11.  A  body  is  thrown  horizontally,   with  an  initial  velocity  of  100  ft. 
per  second,  from  the  top  of  a  tower  150  ft.  high.     At  what  distance  from 
the  tower  will  the  body  strike  the  ground? 

12.  An  arrow  is  shot  vertically  up  with  a  velocity  of  42  m.  per  sec. 
(a)  How  long  will  it  rise?     (b)  How  high  will  it  rise? 


120  DYNAMICS 

13.  A  ball  is  thrown  upward  at  an  angle  of  30°  with  the  horizontal,  with 
an  initial  velocity  of  35  m.  per  second,  (a)  What  is  the  time  of  its  flight? 
(b)  How  high  does  it  rise?  (c)  How  far  from  the  starting  point  does  it  strike 
the  ground? 

SUGGESTION. — Resolve  the  initial  velocity  into  horizontal  and  vertical 
components.  The  first  component  is  constant;  the  second  is  affected  by 
gravity,  just  as  it  would  be  if  the  first  component  did  not  exist. 

II.  NEWTON'S  LAWS  OF  MOTION 

110.  The  General  Laws  of  Dynamics.  —  In  the  preceding 
pages  we  have  considered  the  arithmetical  and  geometrical 
relations    involved    in    certain    types    of   motion.     These 
relations  may  be  termed  "laws  of  motion";  but  they  are 
mathematical  rather   than  physical   laws.     "The  laws  of 
motion,"  preeminently  so  called,  are  the  three  general  laws 
of  dynamics,   which,   taken   together,  completely  express 
the  relations  between  mass,  force,  and  motion.     They  were 
first   stated  in  their  present  form  by  Sir  Isaac  Newton 
in  his  "Principia"  (1687),  and  are  universally  known  as 
Newton's  first,  second,  and  third  laws  of  motion. 

111.  The  First  Law  of  Motion,  or  the  Law  of  Inertia.  - 

Every  body  continues  in  its  state  of  rest  or  of  uniform  motion 
in  a  straight  line,  except  in  so  far  as  it  is  compelled  by  exter- 
nal force  to  change  that  state. 

This  law  has  already  been  considered  (Art.  8);  but  the 
pupil  is  now  in  position  to  understand  it  more  fully.  The 
law  asserts  that  a  body  at  rest  would  remain  at  rest  if  no 
force  acted  upon  it.  This  is  an  ideal  case  impossible  of 
realization,  since  gravity,  at  least,  always  acts.  Assuming 
the  truth  of  the  law,  we  conclude  that,  if  a  body  at  rest 
remains  at  rest,  all  the  forces  acting  upon  it  completely 
neutralize  one  another's  tendency  to  move  the  body;  and 


NEWTON'S    LAWS    OF    MOTION  121 

these  forces,  by  definition,  constitute  a  set  of  balanced 
forces  (Arts.  10  and  65). 

The  law  asserts  further  that  a  body  in  motion  would 
continue  in  motion  with  constant  speed  in  a  straight  line, 
if  no  force  acted  upon  it.  This  is  also  an  ideal  case,  and 
impossible  of  realization  for  the  same  reason  as  before. 
Again  assuming  the  truth  of  the  law,  we  conclude  that,  if 
a  body  in  motion  continues  with  constant  speed  in  a  straight 
line,  all  the  forces  acting  upon  it  completely  neutralize 
one  another's  tendency  to  change  the  motion  of  the  body; 
and  these  forces,  by  definition,  also  constitute  a  set  of  bal- 
anced forces.  To  illustrate:  The  force  necessary  to  pull 
a  load  at  a  uniform  rate  over  a  level  surface  is,  by  the  law 
of  inertia,  equal  to  the  sum  of  all  the  resisting  forces  of  fric- 
tion; and  the  resultant  of  all  the  forces  acting  on  the  load, 
including  friction,  is  zero.  When  mud  flies  from  the  wheels 
of  a  carriage,  or  water  from  a  rapidly  revolving  grindstone, 
or  a  stone  from  a  sling,  or  an  automobile  overturns  in  round- 
ing a  corner  too  quickly,  the  phenomenon  is  nothing  more 
than  an  exhibition  of  this  universal  tendency  of  moving 
matter  to  continue  in  its  existing  direction  of  motion. 
The  body  inevitably  "flies  off  at  a  tangent"  to  its  curved 
path  at  the  instant  when  the  unbalanced  force  that  was 
causing  the  change  of  direction  ceases  or  becomes  inade- 
quate to  the  duty  expected  of  it. 

The  law  of  inertia,  as  will  presently  be  shown,  follows  as 
a  corollary,  or  special  case,  from  the  second  law  of  motion. 
Hence  whatever  experimental  evidence  there  may  be  of 
the  truth  of  the  second  law  holds  equally  for  the  first. 

112.  Relation  Between  Force  and  Acceleration,  with  a 
Given  Mass.  —  The  second  law  of  motion,  in  its  usual  form, 
is  better  understood  from  a  previous  study  of  two  other 


122  DYNAMICS 

laws,  which  are  together  equivalent  to  it.     These  we  shall 
designate  as  laws  2 a  and  26. 

20.  The  acceleration  of  a  given  mass  is  proportional  to 
the  resultant  force  acting  upon  it,  and  is  in  the  direction  of 
this  force. 

The  law  asserts  that  the  acceleration  of  a  body  is  due  to 
the  unbalanced  or  resultant  force  acting  upon  it,  and  that 
the  two  vary  in  the  same  ratio.  This  is  illustrated  by  the 
motion  of  a  sphere  on  a  plane,  when  inclined  at  different 
angles.  The  resultant  or  accelerating  force  upon  the  sphere 
is  the  component  of  its  weight  whose  direction  is  parallel 
to  the  plane  (Art.  74).  This  component  is  doubled  when 
the  height  of  the  plane  (BC,  Fig.  52)  is  doubled;  and  the 
ball  then  goes  twice  as  far  in  the  same  time,  showing  that 
its  acceleration  has  been  doubled.  With  any  two  adjust- 
ments of  the  plane,  the  heights,  the  unbalanced  forces, 
and  the  accelerations  are  all  in  the  same  ratio.  In  firing 
a  cannon  we  have  an  excellent  example  of  an  enormous 
unbalanced  force  and  its  effect.  A  twelve-inch  cannon  50 
ft.  long  launches  a  projectile  with  a  muzzle  velocity  of  3000 
ft.  per  second.  This  velocity  is  imparted  in  about  one 
hundredth  of  a  second,  and  the  average  acceleration  is 
approximately  50  miles  per  second  per  second.  This 
necessitates  an  enormous  accelerating  force,  which,  with 
the  standard  charge  of  powder  and  projectile,  amounts 
to  no  less  than  3,600,000  Ib.  (1800  tons)  at  the  breech  of 
the  gun. 

If  the  resultant  force  upon  a  body  varies  during  its  motion,  the 
acceleration  varies  in  the  same  ratio;  if  the  resultant  force  remains 
constant,  the  acceleration  is  also  constant.  The  motion  of  a  sphere 
on  an  inclined  plane  is  uniformly  accelerated  (Art.  105)  because  the 
accelerating  component  of  its  weight  remains  constant  for  all  posi- 
tions on  the  plane.  The  resultant  force  upon  a  freely  falling  body  is 


NEWTON'S    LAWS    OF    MOTION  123 

its  whole  weight,  if  the  resistance  of  the  air  is  negligible;  and  this 
is  a  constant  force,  hence  the  acceleration  is  constant.  If  the  body 
is  light  and  has  a  relatively  large  surface,  as  a  leaf 
or  a  sheet  of  paper,  the  resistance  of  the  air  is 
relatively  large,  and  the  resultant  force  upon  the 
body  as  it  falls  is  the  difference  between  its  weight 
and  the  resistance  of  the  air.  Hence  the  accele- 
ration in  such  cases  is  less  than  that  of  compact, 
heavy  bodies.  Moreover,  the  resistance  of  the 
air  rapidly  increases  with  the  velocity,  as  every 
bicycle  rider  knows  from  experience;  hence  as  a 
body  falls,  especially  a  light  one,  the  resultant 
force  upon  it  steadily  diminishes,  and  may  even 
become  zero  (the  resistance  of  the  air  having  be- 
come equal  to  the  weight  of  the  body).  The  rate 

of  fall  under  such   conditions  is  constant,  as  in 

^,  ,  .      ,  i          /r.  N  FIG.    112. — An   Ex- 

the  case  of  a  rain-drop  or  a  parachute  (r  ig.  112)      treme  Case  of  Air 

after  it  has  fallen  a  few  hundred  feet.  Resistance 

The  law  holds  for  negative  as  well  as  for  positive  accel- 
eration. The  rate  at  which  the  speed  of  a  body  decreases 
is  in  proportion  to  the  resultant  force  acting  opposite  to  the 
direction  in  which  the  body  is  moving.  It  takes  a  greater 
retarding  force  to  stop  a  car  more  quickly,  just  as  it  takes 
a  greater  accelerating  force  to  start  it  more  quickly. 

The  law  further  asserts  that  the  acceleration  is  in  the 
direction  of  the  force.  The  motion  and  the  resultant  force 
may  be  in  the  same  direction,  in  opposite  directions,  or 
at  any  angle  with  each  other;  but  the  acceleration  and  the 
force  are  always  in  the  same  line  and  in  the  same  direction 
along  that  line.  This  is  admirably  illustrated  by  the 
motion  of  a  heavy  body  in  the  air.  The  resultant  force 
is  the  weight  of  the  body,  and  the  acceleration  is  always  in 
the  direction  of  gravity  (vertically  downward),  regardless 
of  the  direction  in  which  the  body  may  be  moving  (Art. 
109,  third  case).  In  all  cases  the  accelerated  component  of 


124  DYNAMICS 

the  motion  of  a  body  is  in  the  direction  of  the  resultant 
force  upon  it.  In  addition  to  this  the  body  may  have  a 
constant  component  of  motion  (previously  imparted)  in 
any  direction. 

The  first  law  of  motion  is  a  corollary  to  the  above;  for, 
since  force  and  acceleration  are  proportional,  when  the 
force  is  zero  the  acceleration  is  also  zero,  and,  with  zero 
acceleration,  motion  remains  unchanged  both  in  speed  and 
direction. 

113.  Relation  between  Force  and  Mass,  with  a  Given 
Acceleration.  —  The  second  law  referred  to  at  the  begin- 
ning of  the  preceding  article  is  as  follows: 

2b.  The  resultant  force  necessary  to  produce  a  given  accel- 
eration is  proportional  to  the  mass  of  the  body  upon  which 
the  force  acts. 

In  lifting  and  carrying  bodies,  in  dragging  or  pushing 
them  along,  in  hauling  loads  in  wagons  or  in  cars,  and 
indeed  in  most  cases  of  motion  with  which  we  are  familiar 
through  daily  observation  and  experience,  friction  and 
weight  are  the  chief  hindrances  to  be  overcome;  and  the 
effect  of  mass,  being  thus  obscured,  is  commonly  not  rec- 
ognized at  all.  The  " resultant  force"  of  the  law  is  the 
resultant  of  all  the  forces  acting  on  the  body,  including 
weight  and  friction  (which  are  thus  fully  allowed  for),  and 
is  the  force  that  mass  makes  necessary  whenever  accelera- 
tion takes  place.  If  bodies  had  mass  but  not  weight,  and 
moved  without  friction  (which  is  true  of  the  earth  and  other 
planets  as  bodies  in  space),  the  laws  of  motion  would  still 
be  the  same.  Bearing  these  facts  in  mind,  let  us  see  how 
the  above  law  applies  in  the  following  cases. 

If  a  horse  exerts  a  force  of  150  Ib.  in  drawing  a  loaded  cart  at  a 
uniform  rate  on  a  level  road,  it  is  because  the  friction  opposing  the 


NEWTON'S    LAWS    OF    MOTION  125 

motion  amounts  to  150  lb.;  and  the  resultant  or  accelerating  force 
on  the  load  is  zero.  To  start  the  load  an  additional  force  is  necessary; 
and,  for  a  given  rate  of  starting,  the  amount  of  this  additional  force 
is  determined  solely  by  the  mass  of  the  load  (including  the  cart). 
A  computation,  which  the  pupil  will  presently  be  able  to  make  for 
himself,  shows  that  to  impart  a  velocity  of  3  mi.  per  hour  during  the 
first  10  sec.  (while  the  cart  goes  the  first  22  ft.)  would  require  an 
accelerating  force  of  a  little  more  than  27  lb.  per  ton  of  the  load. 
This,  it  will  be  observed,  is  much  less  than  the  force  ordinarily 
required  to  overcome  friction.  But  to  impart  a  velocity  of  3000  ft. 
per  second  to  a  thousand-pound  projectile  while  it  is  moving  from  the 
breech  to  the  muzzle  of  a  twelve-inch  gun  requires  an  average 
accelerating  force  of  1400  t.,  or  2800  t.  of  force  per  ton  of  mass,  —  a 
force  in  comparison  with  which  weight  and  friction  are  utterly 
insignificant. 

Further  light  on  the  meaning  of  the  law  is  afforded  by 
retarded  motion.  While  friction  acts  as  the  chief  obstacle 
to  starting  in  ordinary  cases,  and  thus  diverts  the  atten- 
tion from  the  effect  of  mass,  it  directly  aids  stopping. 
Yet  even  when  friction  is  greatly  increased  by  applying 
brakes,  time  is  required  to  stop  a  car  or  other_jvehicle,and 
the  greater  its  mass  the  greater  is  the  difficulty  in  stopping 
it.  The  force  of  gravity  (weight)  is  in  no  wise  responsible 
for  this  difficulty. 

The  pupil  should  impress  his  mind  with  the  idea  of  mass 
by  a  few  experiments  in  quickly  starting  and  stopping 
bodies,  under  such  conditions  that  weight  and  friction 
play  an  unimportant  part,  such  as  setting  in  motion  a 
grindstone  or  a  massive,  well  balanced  wheel,  and  then 
endeavoring  to  stop  it,  or  rapidly  twirling  a  long  pole 
from  side  to  side  in  the  hand.  This  may  seem  to  be 
nothing  more  than  a  study  of  inertfa,  and  that  is  pre- 
cisely what  it  is.  Mass  and  inertia,  as  measurable  quan- 
tities, are  identical. 


126 


DYNAMICS 


The  proportionality  stated  in  the  law  is  admirably  illustrated  by 
falling  bodies.  All  bodies  which  are  not  measurably  retarded  by  the 
resistance  of  the  air,  and  all  bodies  without  exception  in  a  vacuum 
(Fig.  113),  are  equally  accelerated  in  falling.  This  is  readily 
tested  by  dropping  a  pebble  and  a  large  stone  from 
the  same  height  at  the  same  time.  They  reach 
the  ground  at  the  same  instant.  (Try  it.)  In  such 
cases  the  accelerating  force  is  the  whole  weight  of 
the  body,  and  weight  is  proportional  to  mass.  Thus  a 
five-pound  mass,  in  falling,  is  accelerated  by  a  force 
five  times  as  great  as  that  upon  a  falling  mass  of  one 
pound;  hence  the  equal  acceleration.  When  weight 
increases  without  an  increase  of  mass,  as  when  a 
body  is  taken  to  a  place  farther  from  the  equator 
(Art.  15),  its  acceleration  in  falling  is  proportionately 
greater  (Law  20).  Thus  at  the  equator  the  accel- 
eration due  to  gravity  is  978  (cm.  per  second  per 
second),  while  at  either  pole  it  is  983. 

114.   Absolute  Units  of  Force.  —  The  earth's  at- 
traction for  the  different   English  and  metric  units 
"Guine'    md  °^  mass  suPPnes  corresponding   units  of  force  (Art. 
Feather  "Ex-  J7)-     These  are  termed  gravitational  units  of  force, 
periment.        since  they  are  due  to  gravity.     They  vary  slightly 
with  latitude  and  altitude  (Arts.  15  and  17);  but  for 
most    purposes   these   variations   are   unimportant.     For  scientific 
purposes,  however,  an   invariable   unit  of   force  is  necessary;   and 
the   one    chosen   has    the   further   advantage    of    simplifying    the 
statement  of   dynamical  laws   and  formulas.     The  dynamical    or 
absolute   unit  of  force  is  that  force   which,   acting   alone  or  as  a 
resultant    force,    produces    unit    acceleration    in    unit   mass.      The 
absolute  unit  of  force  in  the  metric  system  is  called  the  dyne.      It 
is  variously  defined  in  equivalent  terms  as  follows:  (i)  The  dyne 
is  that  force  which,  acting  on  a  mass  of  one  gram  for  one  second,  pro- 
duces a  change  of  velocity  of  one  centimeter  per  second;  or  (2)  it 
is  that  force  which,  acting  continuously,  imparts  an  acceleration  of 
one  centimeter  per  second  per  second  to  a  mass  of  one  gram;  or  (3) 
it  is  that  force  which,  acting  on  any  mass,  changes  its  momentum  at 
the  rate  of  one  unit  per  second  (mass  being  expressed  in  grams  and 


NEWTON'S    LAWS    OF    MOTION  127 

velocity  in  centimeters  per  second).  (See  Arts.  115  and  116  for 
meaning  of  the  term  momentum.) 

Since  the  weight  of  a  gram  mass  gives  it  an  acceleration  980  times 
as  great  as  a  force  of  one  dyne  would  give  it,  it  follows  (Law  20) 
that  the  gram  weight  is  equal  to  980  dynes.  Strictly  speaking,  the 
weight  of  a  gram  at  sea-level  is  equal  to  978  dynes  at  the  equator,  to 
980  dynes  in  the  latitude  of  New  York,  and  to  983  dynes  at  the  pole 
(Art.  113).  It  is  the  gram  weight  that  varies,  not  the  dyne. 

The  English  absolute  unit  of  force  is  called  the  poundal,  and  is 
denned  as  that  force  which  imparts  to  a  pound  mass  an  acceleration 
of  one  foot  per  second  per  second.  Taking  the  acceleration  of  a 
falling  body  as  32.16  ft.  per  second  per  second,  a  poundal  is  y^T*  of 
a  pound  weight.  (Why?) 

115.  Momentum.  —  Certain  dynamical  relations,  which 
are  presently  to  be  considered,  involve  the  product  of  the 
mass  of  a  body  and  its  velocity.     This  product  is  called 
the  momentum  of  the  body.     It  is  generally  expressed  as 
the  product  of  grams  and  centimeters  per  second,  or  as  the 
product  of  pounds  and  feet  per  second;  but  other  units 
may  be  employed.     No  name  has  been  given  to  any  unit 
of  momentum. 

Since  the  mass  of  a  body  is  constant,  its  momentum 
changes  only  with  a  change  of  velocity.  Hence  the  rate 
at  which  momentum  changes,  is  measured  by  the  product 
of  the  mass  of  the  body  and  the  rate  at  which  its  velocity 
changes;  or,  more  briefly,  rate  of  change  of  momentum  is 
equal  to  the  product  of  mass  and  acceleration.  Let  m  denote 
the  mass  of  a  body,  v  its  velocity,  and  a  its  acceleration; 
then  mv  is  its  momentum,  and  ma  is  the  rate  at  which  its 
momentum  is  changing. 

116.  Newton's  Second  Law,  or  the  Law  of  Accelerated  Motion.  — 
A  force  of  12  dynes  acting  on  a  mass  of  i  g.  gives  it  an  acceleration  of 
1 2  cm.  per  second  per  second  (Law  20).     To  impart  an  equal  accelera- 
tion to  a  mass  of  8  g.  would  require  8  times  as  great  a  force  (Law  26), 


128  DYNAMICS 

or  96  dynes.  The  number  of  dynes  is  thus  equal  to  the  product  of 
the  mass  in  grams  and  the  acceleration  in  centimeters  per  second  per 
second.  This  relation  is  general.  Expressed  as  a  formula  it  is  — 

/  =  wa,  (13) 

the  units  in  which  the  quantities  are  measured  being  the  dyne,  the 
gram  mass,  and  the  centimeter  per  second  per  second;  or  the  poundal, 
the  pound  mass,  and  the  foot  per  second  per  second. 

Since  a  force  in  dynes  is  changed  to  grams  by  dividing  by  g  ( =  980 
cm.  per  second  per  second)  and  a  force  in  poundals  to  pounds  by 
dividing  by  g  (=  32.16  ft.  per  second  per  second),  the  formula 

becomes 

ma 
f  =  y'  (14) 

when  the  force  is  measured  either  in  grams  or  pounds  and  the  other 
quantities  in  the  corresponding  units. 

The  product  ma  in  either  formula  is  the  rate  of  change  of  momen- 
tum (Art.  115);  hence  the  rate  of  change  of  the  momentum  of  a  body 
is  equal  to  the  resultant  force  in  dynes  or  poundals  acting  upon  it, 
and  is  proportional  to  the  resultant  force  in  grams  or  pounds.  Adding 
to  this  the  fact  that  the  change  of  momentum  is  in  the  direction  of 
the  acceleration  and  that  the  acceleration  is  in  the  direction  of  the 
force,  we  have  — 

The  Second  Law  of  Motion:  Rate  of  change  of  momentum  is  propor- 
tional to  the  resultant  force,  and  takes  place  in  the  direction  of  the  force. 

117.  ^Impulse.     The    Second  Law    of   Motion   Restated. — The 
change  of  velocity  produced  in  a  given  mass  by  an  unbalanced  force 
is  proportional  to  the  time  during  which  the  force  acts,  as  well  as  to 
the  magnitude  of  the  force.   (Why?)   Con- 
sequently, if  a  force  acts  only  for  an  in- 
stant, the  motion  produced  will  be  slight, 
unless  the  force  itself  is  very  great.     This 
is  especially  true  if  the  body  has  consider- 
able mass.     The  force  exerted  by  a  bullet 
in  penetrating  a  board  is  a  good  example. 
The  time  is  so  extremely  brief  that  a  bullet 
shot   through  a  board  standing  on  edge, 
will  not  overturn  it,  although  it  can  easily 
FIG.  114-  be  overturned  with  a  finger. 


NEWTON'S    LAWS    OF    MOTION  129 

The  importance  of  time  in  the  action  of  a  force  is  well  shown  by 
a  simple  experiment  with  a  small  coin  placed  on  a  small  card.  The 
friction  between  them  is  sufficient  to  impart  the  motion  of  the  card 
to  the  coin  when  the  card  is  moved  slowly  about;  but,  when  it  is 
started  suddenly,  the  coin  is  left  behind.  This  is  neatly  shown  by 
snapping  the  card  from  under  it  (Fig.  114).  If  the  blow  is  aimed 
successfully,  the  coin  will  be  left  at  rest  on  the  finger,  friction  being 
insufficient  to  impart  appreciable  motion  to  it  in  so  short  a  time. 
(Try  it.) 

Let  /  denote  the  force  in  dynes  or  poundals  acting  on  a  mass  m, 
a  the  acceleration  due  to  the  force,  and  v  the  velocity  imparted  in  t 
seconds;  then 

/  =  ma, 

and  v  =  at. 

Eliminating  a,  ft  =  mv.  (15) 

That  is,  the  momentum  imparted,  or  the  change  of  momentum, 
is  equal  to  the  product  of  the  force  and  the  time  during  which  it 
acts.  The  product  //  is  called  the  impulse  of  the  force.  Hence  one 
way  of  stating  the  second  law  of  motion  is  as  follows:  The  change 
of  momentum  of  a  body  is  equal  to  the  impulse  which  produces  it,  and 
is  ~in  the  direction  of  the  impulse. 

118.  The  Third  Law  of  Motion,  or  the  Law  of  Mutual 
Action.  —  Newton's  third  law  of  motion  is  as  follows :  To 
every  action  there  is  an  equal  and  opposite  reaction;  or,  when- 
ever one  body  exerts  a  force  on  another,  the  other  exerts  an  equal 
and  opposite  force  on  the  first. 

This  law  is  stated  and  discussed  at  some  length  in  Art. 
12.  It  asserts  that  every  force  is  one  of  a  pair  of  equal  and 
opposite  forces,  exerted  by  two  bodies  or  by  two  parts 
of  the  same  body  on  each  other.  The  two  forces  involved 
in  "  action  and  reaction  "  can  not,  under  any  circumstances, 
balance  each  other,  since  they  act  upon  different  bodies  or 
portions  of  matter;  but  either  or  both  may  be  balanced 
by  other  forces  acting  at  the  same  time.  For  example, 
the  pressure  exerted  by  a  bat  against  a  ball  in  striking  it 


130  DYNAMICS 

is  unbalanced,  and  imparts  motion  to  the  ball.  The 
reaction  of  the  ball  against  the  bat  is  also  unbalanced,  and 
checks  the  motion  of  the  bat.  When  a  piece  of  iron,  placed 
on  an  anvil,  is  struck  with  a  hammer,  the  downward  blow 
of  the  hammer  is  balanced  by  the  equal  upward  pressure 
of  the  anvil,  both  acting  on  the  piece  of  iron;  hence  the  iron 
remains  at  rest.  When  a  person  jumps  from  a  boat,  the 
reaction  on  the  boat  is  unbalanced  and  pushes  the  boat  in 
the  opposite  direction  from  that  in  which  the  person 
jumps;  but,  in  jumping  from  a  rock,  the  reaction  against 
the  rock  is  balanced  by  friction  between  it  and  the  ground, 
and  it  remains  at  rest. 

The  equal  forces  exerted  by  two  interacting  bodies  upon  each  other 
necessarily  act  for  equal  times;  hence  the  impulses  of  the  two  forces 
are  equal,  and,  if  the  bodies  are  free  to  move,  equal  changes  of  momen- 
tum are  produced  in  them.  Let  m\  and  mz  denote  the  masses  of  the 
two  bodies,  and  v\  and  %  their  respective  velocities  imparted  by  mutual 
action,  the  bodies  being  initially  at  rest  and  free  to  move;  then  m\vi  = 
m^i  and  m\  :  m%  ::  ^  •  »i,  i-e-  the  velocities  imparted  to  the  bodies 
are  inversely  proportional  to  their  masses.  Thus  if  a  man  jumps 
from  a  boat  whose  mass  is  three  times  that  of  his  own  body,  the 
boat  is  pushed  back  with  a  velocity  one  third  as  great  as  the  forward 
velocity  of  the  man.  When  a  moving  body  strikes  a  body  at  rest, 
and  their  mutual  actions  are  unbalanced,  the  one  loses  as  much 
momentum  as  the  other  gains. 

119.  Scope  of  the  Laws.  —  The  motion  of  all  matter,  animate  as 
well  as  inanimate,  is  in  full  accord  with  Newton's  laws  of  motion. 
It  is  a  common  error  to  regard  the  motions  of  animals  and  self- 
propelling  machines  as  exceptions  to  the  law  of  inertia,  because  they 
"make  themselves  go."  The  motion  of  a  train  presents  no  difficulty 
on  this  score  so  long  as  we  consider  only  the  cars,  for  the  external 
force  is  easily  identified;  it  is  the  pull  of  the  engine.  But  what  pulls 
or  pushes  the  engine?  We  get  a  clew  to  the  answer  when  we  see  the 
drive-wheels  of  an  engine  slip  and  spin  round,  as  sometimes  happens 
in  starting  a  heavily  loaded  train.  Sharp  sand  sprinkled  on  the  rails 


NEWTON'S    LAWS    OF    MOTION 


(from  the  sand  box  above  the  boiler)  remedies  the  difficulty  by 
increasing  the  friction  between  wheel  and  rail.  Thus  while  friction 
between  the  car-wheels  and  the  rails  is  a  hindrance  to  the  motion 
of  the  train,  between  the  drive-wheels  of  the  engine  and  the  rails  it 
is  a  necessity.  The  drive-wheels  exert  a  backward  thrust  on  the  rails; 
the  rails  an  equal  forward  thrust  (reaction)  on  the  wheels.  This 
forward  reaction  on  the  drive-wheels  is  the  external  force  which  not 
only  enables  the  engine  to  "  move  itself  "  but  also  to  pull  the  train. 
Newton's  three  laws,  as  already  stated,  completely  express  the 
relations  between  mass,  force,  and  motion.  Starting  with  these 
laws  as  given  data,  the  mathematical  physicist  can  derive  the  laws 
of  motion  in  all  special  cases,  such  as  the  laws  of  curvilinear  motion 
(Art.  122),  the  laws  of  planetary  motion  (Art.  125),  the  laws  of  the 
pendulum  (Art.  128),  the  laws  of  machines  (Arts.  138-147),  the  laws 
of  vibration  of  strings  in  music  (Art.  280),  etc.  Such  work,  how- 
ever, lies  almost  wholly  in  the  field  of  advanced  physics. 

120.  Galileo  and  Newton.  —  The  creation  of  the  science  of  dynam- 
ics is  due  to  Galileo  Galilei  (1564-1642),  an  Italian  mathematician 
and  physicist.  "The  first  experiments  which  Galileo  made  while 
he  was  a  young  professor  at  Pisa  were  deci- 
dedly dramatic.  At  that  time  the  doctrine 
that  the  rate  at  which  a  body  falls  depends 
upon  its.  weight  was  generally  accepted  as 
true,  merely  on  the  authority  of  Aristotle.  It 
was  even  held  that  the  acceleration  varies 
as  the  weight.  Prior  to  Galileo  it  had  not 
occurred  to  any  one  actually  to  try  the  experi- 
ment. The  young  professor's  tests  went  con- 
trary to  the  doctrine  held  for  two  thousand 
years.  Allowing  for  the  resistance  of  the  air, 
he  found  that  all  bodies  fell  at  the  same  rate, 
and  that  the  distance  passed  over  varied  as 
the  square  of  the  time.  With  all  the  enthu- 
siasm, courage,  and  imprudence  of  youth,  the 
experimenter  proclaimed  that  Aristotle,  at  that 
time  believed  by  nearly  every  one  to  be  verbally  inspired,  was 
wrong.  Galileo  met  with  opposition,  but  he  decided  to  give  his 
opponents  ocular  proof.  It  seems  almost  as  if  nature  had  resorted 


FIG.  115.  —  Leaning 
Tower  of  Pisa. 


132  DYNAMICS 

to  an  extraordinary  freak  to  furnish  Galileo,  at  this  critical  moment 
in  the  history  of  science,  with  an  unusual  convenience  for  his  pub- 
lic demonstration.  Yonder  tower  of  Pisa  had  bent  over  to  facilitate 
experimentation,  from  its  top,  on  falling  bodies.  One  morning, 
before  the  assembled  university,  he  ascended  the  leaning  tower,  and 
allowed  a  one-pound  shot  and  a  one-hundred-pound  shot  to  drop 
together.  The  multitude  saw  the  balls  start  together,  fall  together, 
and  heard  them  strike  the  ground  together.  Some  were  convinced, 
others  returned  to  their  rooms,  consulted  Aristotle,  and,  distrusting 
the  evidence  of  their  senses,  declared  continued  allegiance  to  his 
doctrine."  (Cajori's  "History  of  Physics.")  Galileo  explained  the 
motion  of  a  projectile,  and  thus,  in  effect,  discovered  the  first  and 
second  laws  of  motion.  He  determined  the  laws  of  the  pendulum 
(Art.  128),  and  suggested  its  use  in  measuring  time. 

The  name  of  Sir  Isaac  Newton  (1642-1727),  an  Englishman,  stands 
preeminent  in  the  history  of  science.  As  mathematician,  astronomer, 
and  physicist,  he  made  invaluable  contributions  to  the  progress  of 
knowledge.  He  formulated  the  general  laws  of  dynamics  which 
bear  his  name,  and  applied  them  with  unexampled  skill  to  the  motions 
of  the  heavenly  bodies  (Art.  125). 

PROBLEMS 

1.  Why  does  a  falling  body,  on  striking  the  earth,  exert  a  pressure 
in  excess  of  that  due  to  its  weight?     Would  the  pressure  be  the  same 
whether  the  ground  was  hard  or  soft? 

2.  A  bullet  fired  through  a  plate  glass  window  will  often  make  a 
smooth  hole  without  cracking  the  glass.     Explain. 

3.  A  nail  can  be  driven  by  striking  it  with  a  hammer,  but  not  by  press- 
ing the  hammer  steadily  against  it.     Explain. 

4.  Gravity  upon  the  moon  is  one  sixth  as  great  as  upon  the  earth. 
Compute  the  acceleration  of  a  falling  body  upon  the  moon. 

5.  Gravity  upon  the  sun  is  27.6  times  as  great  as  upon  the  earth. 
Compute  the  acceleration  of  a  falling  body  upon  the  sun. 

6.  How  far  would  a  body  fall  during  the  first  second    (a)  upon  the 
moon?  (&)  upon  the  sun? 

7.  (a)  Would  the  mass  of  a  given  body  be  the  same  upon  the  sun  01 
the  moon  as  upon  the  earth?     (6)  Would  its  inertia  be  the  same? 


NEWTON'S   LAWS   OF   MOTION  133 

8.  Would  it  take  less  powder  to  fire  a  cannon  ball  with  a  given  veloc- 
ity upon  the  moon  than  it  would  upon  the  earth? 

9.  Is  it  harder  for  horses  to  start  a  loaded  wagon  or  to  keep  it  in  uni- 
form motion?     Give  reasons. 

10.  Why  does  a  ball  player  move  his  hands  quickly  backward  in  the  act 
of  catching  a  swift  ball? 

11.  Is  mass,  weight,  or  friction  principally  responsible  for  the  difficulty 
experienced  in  quickly  getting  up  speed  on  a  bicycle? 

12.  State  the  meaning  of  the  formula  ft  =  mv  when  the  body  has  an 
initial  velocity  and  /  is  in  the  direction  of  motion;   also  when  /  is  opposite 
to  the  direction  of  motion. 

13.  How  is  the  weight  of  a  body  in  dynes  obtained  from  its  weight  in 
grams?     How  is  its  weight  in  poundals  obtained  from  its  weight  in  pounds? 

14.  Account  for  the  motion  of  a  revolving  lawn  sprinkler. 

15.  How  does  the  mutual  action  between  the  front  wheel  of  a  bicycle  and 
the  ground  differ  from  that  between    the  rear  wheel  and   the  ground? 
Explain. 

16.  How  does  it  follow  from  the  second  law  of  motion  that  an  unbalanced 
force,  however  small,  acting  on  any  mass,  however  great,  will  move  it  or 
change  its  existing  motion? 

17.  If  equal  forces  impart  equal  accelerations  to  two  bodies,  how  do  the 
masses  of  the  bodies  compare,  and  why? 

18.  Does  a  horse  pull  harder  upon  a  wagon  in  drawing  it  than  the  wagon 
does  on  the  horse?     Explain. 

19.  Why  does  an  elevator  cable  pull  more  than  the  weight  of  the  car 
and  occupants  while  gaining  velocity  going  up,  and  less  than  the  weight  of 
the  car  and  occupants  while  gaining  velocity  going  down?     Is  the  pull 
greater  or  less  than  the  weight,  and  why,  while  losing  velocity  going  up  and 
while  losing  velocity  going  down? 

20.  How  is  the  momentum  which  is  produced  by  a  given  impulse  affected 
by  the  mass  of   the  body  acted  upon?     How  is  the  velocity  imparted 
affected  by  the  mass  of  the  body? 

21.  If  equal  impulses  impart  equal  velocities  to  two  bodies,  how  do  the 
masses  of  the  bodies  compare,  and  why? 

22.  Is  the  air  necessary  for  the  flight  of  a  bird?     Why  or  why  not? 
Discuss  as  definitely  as  you  can  the  mechanics  of  a  bird's  flight. 


134  DYNAMICS 

23.  Why  does  a  person  slip  in  trying  to  start,  stop,  or  turn  quickly  on 
ice,  with  ordinary  shoes  on  the  feet?     How  do  skates  remedy  the  difficulty? 

24.  Discuss  and  compare  the  mechanics  of  the  standing  broad  jump  and 
the  running  broad  jump. 

25.  Why  is  a  locomotive  built  so  that  its  drive-wheels  sustain  as  much 
of  its  weight  as  is  possible? 

26.  Why  do  the  drive-wheels  of  an  engine  sometimes  slip,  spinning 
round  and  round,  while  the  other  wheels  and  the  wheels  of  the  cars  never  do? 

27.  A  bullet  is  fired  from  a  rifle  with  a  muzzle  velocity  of  2000  ft.  per 
second.     The  bullet  weighs  £  oz.  and  the  rifle  10  Ib.     What  is  the  velocity 
of  the  rifle  in  the  recoil?     How  is  this  velocity  imparted  to  the  rifle? 

28.  Two  boys,  A  and  B,  are  pulling  upon  the  ends  of  a  rope.     A  pulls  B 
along.     Is  he  exerting  a  greater  pull  than  B?     Explain. 

29.  Why  does  stamping  remove  mud  from  the  shoes?    Why  can  not  all 
of  the  mud  be  removed  in  this  way? 

30.  The  handle  can  be  tightened  in  the  head  of  an  ax  by  striking  the 
end  of  the  handle  against  a  log,  or  by  holding  the  ax  at  rest  and  striking 
the  end  of  the  handle  with  a  hammer.     Explain. 

31.  A  plane  is  inclined  so  that  i  of  the  weight  of  a  sphere  placed  on  it  is 
effective  in  causing  acceleration  down  the  plane.     How  far  will  the  sphere 
roll  in  2.5  seconds? 

32.  A  falling  body  weighs  100  g.     What  is  its  acceleration  at  the  instant 
when  the  resistance  of  the  air  against  it  is  25  g.? 

33.  A  block  weighing  1000  g.  slides  down  a  plane  200  cm.  long,  inclined 
to  a  height  of  120  cm.     The  resistance  of  friction  is  275  g.     Find  (a)  the 
accelerating  force,  (b)  the  acceleration,  (c)  the  rate  of  gain  of  momentum. 

34.  The  mass  of  a  wagon  and  its  load  is  3  tons,  and  the  resistance  of 
friction  is  250  Ib.     In  what  time  and  in  what  distance  will  it  come  to  rest 
if  it  is  moving  at  the  rate  of  3  mi.  per  hour  when  the  horses  stop  pulling? 

35.  A  body  weighing  5  Ib.  is  projected  vertically  upward  by  a  constant 
force/,  acting  through  a  distance  of  3  ft.;  and  it  rises  100  ft.  higher  before 
coming  to  rest.     Find  /. 

SUGGESTION.  —  Find  the  velocity  necessary  to  enable  a  body  to  rise  100  ft. 
against  gravity;  the  acceleration  necessary  to  impart  this  velocity  in  3  ft.; 
the  resultant  force  necessary  to  produce  this  acceleration  in  a  mass  of  5  Ib.; 
and,  finally,  the  whole  upward  force  /. 


CURVILINEAR   MOTION 


135 


FIG.  116.  —  Path  of  a  Projectile. 


III.  THE  LAWS  OF  MOTION  IN  SPECIAL  CASES 

121.  Curvilinear  Motion.  —  Change  of  speed  is  due  to 
unbalanced  force  in  the  line  of  motion;  change  of  direc- 
tion, to  unbalanced  force  at  right  angles  to  the  line  of 
motion.  If  the  result- 
ant force  upon  a  body 
is  oblique  to  the  line 
of  motion,  it  is  equiva- 
lent to  two  compo- 
nents, one  of  which  is  jv 
in  the  line  of  motion 
and  the  other  at  right 
angles  to  it.  The  weight 
of  a  projectile  is  an  excellent  example.  Let  LM  N  (Fig.  1 16) 
represent  the  path  of  a  projectile.  At  L,  in  rising,  the  com- 
ponent Toi  the  weight  causes  decrease  of  speed,  and  the  com- 
ponent /  causes  change  of  direction.  At  the  highest  point 
M,  the  whole  weight/ causes  change  of  direction.  At  N,  in 
falling,  the  component  T  causes  increase  of  speed,  and  the 

component  /  causes  change 
of  direction.  The  path  every- 
where curves  in  the  direction 
of  the  component  /,  and  the 
curvature  is  greatest  at  M, 
where  /  is  greatest.  The 
line  of  motion  at  any  point  of 
a  curved  path  is  the  straight 
line  tangent  to  the  curve  at 
that  point,  and  the  compo- 
nent force  in  the  line  of  mo- 
tion is  called  the  tangential  force.  The  component  force 
at  right  angles  to  the  path  is  called  the  centripetal  force, 


136  DYNAMICS 

because  it  acts  toward  the  center  of  the  curved  path  (from 
the  Latin  centrum,  center,  and  peter e,  to  seek). 

Since  change  of  direction  of  motion  is  due  to  unbalanced 
centripetal  force,  the  change  of  direction  ceases  at  the 
instant  when  the  centripetal  force  ceases;  and  the  body, 
from  that  instant,  continues  in  a  straight  line,  as  the  re- 
sult of  its  inertia.  This  is  readily  shown  by  means  of  a 
ball  fastened  to  the  end  of  a  string.  When  the  ball  is  rolled 
in  a  circle  on  the  top  of  a  table  or  on  the  floor  and  suddenly 
released,  as  at  A  (Fig.  117),  it  continues  in  the  direction  in 
which  it  was  moving  at  the  instant  of  release.  Similarly, 
when  a  stone  is  whirled  in  a  sling  and  released,  it  "  flies 
off  at  a  tangent."  The  tendency  of  bodies  to  leave  a 
curved  path  is  commonly  called  a  centrifugal  tendency 
(from  the  Latin  centrum,  audfugere,  to  flee,  i.e.  fleeing  from 
the  center);  but  it  is  nothing  else  than  the  universal  ten- 
dency of  moving  bodies  to  maintain  a  straight  course,  as 
expressed  in  the  first  law  of  motion  (Art.  in).  It  should 
be  noted  that  the  centrifugal  motion  which  occurs  when 
the  centripetal  force  ceases  is  not  outward  along  a  radius, 
but  outward  along  a  tangent  to  the  curved  path. 

122.  Laws  of  Centripetal  Force.  —  If  a  ball  is  suspended 
by  a  string  from  a  fixed  support  and  started  in  a  horizon- 
tal circle  (Fig.  118),  it  will  continue  to  revolve  in  a  slowly 
diminishing  circle  (more  accurately  a  spiral)  for  several 
minutes.  The  decrease  in  the  size  of  the  circle  is  due  to 
friction,  chiefly  of  the  air,  and  may  be  disregarded.  If  all 
friction  could  be  removed,  the  motion  in  a  circle  would 
continue  indefinitely.  Disregarding  friction,  the  ball  is 
acted  upon  by  two  forces;  namely,  its  weight,  W,  and  the 
tension,  T,  of  the  cord.  The  vertical  component  of  the 
tension,  v,  is  equal  to  W  and  balances  it;  the  horizontal 


CURVILINEAR   MOTION 


137 


FIG.  118. 


component  /  is  unbalanced,  and  is  directed  toward  the 
center  of  the  circle.  The  centripetal  force  /  holds  the 
ball  in  a  circular  path,  but 
has  no  effect  on  its  speed. 
If  the  ball  is  whirled  more 
and  more  rapidly,  the  string 
being  held  in  the  hand,  the 
circle  in  which  it  revolves 
grows  larger,  and  the  cord 
becomes  more  nearly  horizon- 
tal (Fig.  119).  The  reason 
for  this  behavior  is  that  a 
greater  speed  is  accompanied 
by  a  more  rapid  change  of 
direction,  and  this  necessi- 
tates a  greater  centripetal  force.  The  ball  consequently 
moves  out  until  the  centripetal  force  has  increased  to  the 

required  value. 

It  would  be  difficult  from  such  an 

experiment  to  determine  the  exact 
relations  among  the  quantities 
involved  in  curvilinear  motion, 
since  three  of  them  vary  at  the 
same  time;  namely,  the  centripe- 
tal force,  the  radius  of  curvature, 
and  the  velocity.  It  can  be  shown, 
however,  both  by  experiment  and 
by  mathematical  analysis,  that 
the  centripetal  force  is  always  pro- 
portional to  the  square  of  the 
velocity,  fl2,  when  the  radius  is  con- 
stant, and  inversely  proportional 
to  the  radius,  r,  when  the  velocity 
is  constant.  If  in  the  above  ex- 
periment a  heavier  ball  were  used,  v  and  /  would  change  in  the 
same  ratio  as  the  weight,  and  hence  also  in  the  same  ratio  as  the 


FIG.  119. 


138  DYNAMICS 

mass  of  the  ball.  This  illustrates  the  general  law  that,  for  a  given 
velocity  and  radius  of  curvature,  the  centripetal  force  is  proportional 
to  the  mass  of  the  body.  When  the  centripetal  force  is  expressed 
in  dynes  or  poundals  and  the  other  quantities  in  the  corresponding 
units,  the  above  laws  give  the  formula  — 

/  =  — -•     (Equation  for  centripetal  force.)  (16) 

It  can  be  shown  that,  for  uniform  speed  in  a  circle,  —is  the  acceler- 
ation toward  the  center  which  results  in  change  of  direction;  hence 
the  above  formula  is  a  special  case  of  the  general  formula  /  =  ma. 

123.   Illustrations  and  Applications  of  the  Laws.  —  In 

all  cases  of  curvilinear  motion  upon  the  earth's  surface,  the 
moving  body  exerts  an  outward  thrust  against  the  ground 
or  other  support,  and  turning  is  effected  by  the  inward  reac- 
tion of  the  supporting  surface.  The  necessity  for  this  out- 
ward thrust  and  inward  reaction  to  accomplish  turning  is 
plainly  shown  in  cases  where  it  is  insufficient  to  meet  the 
demands  made  upon  it,  as  when  a  bicycle  rider  attempts 
to  turn  quickly  on  a  wet  pavement  and  the  wheel  slips  out- 
ward from  under  him.  Un- 
der like  circumstances,  an 
automobile  "  skids,"  sliding 
sidewise  toward  the  outside 
of  the  curve. 

The  outward  thrust  of  cars 
in  turning  a  curve  comes  upon 
the  outer  rail;  and  for  heavy 
trains  moving  at  high  speed, 
this  thrust  is  enormous,  some- 
times, indeed,  so  great  as  to  pull 
out  or  shear  off  the  spikes  which 
hold  the  rail  in  place,  causing 
disastrous  wrecks.  To  reduce 

this  dangerous  thrust  as  much  as  possible,  the  road-bed  of  a  track 
is  always  raised  on  the  outside  of  a  curve  (Fig.  120).  The  correct 


FIG.  1 20.  —  The  Resultant  Force  on  the 
Car  is  toward  the  Center  of  the  Curve. 


CURVILINEAR    MOTION  139 

inclination  would  be  such  as  to  bring  the  road-bed  at  right  angles 
to  the  resultant  force  between  the  car  and  the  rails.  But  since  the 
outward  thrust  of  the  car  varies  as  the  square  of  its  velocity,  and  the 
velocity  is  not  always  the  same,  the  inclination  adopted  is,  at  best, 
a  compromise.  In  the  figure  W  represents  the  weight  of  the  car,  P 
the  entire  reaction  of  the  track  due  to  the  weight  and  the  outward 
thrust  of  the  car  (taken  as  acting  at  the  center  of  gravity),  and  / 
the  resultant  or  centripetal  force  upon  the  car. 

Centrifugal  motion  is  usefully  applied  in  separating  one  sub- 
stance from  another,  generally  a  liquid  from  a  solid.  The  solid  is 
placed  in  a  perforated  cylindrical  vessel,  which  is  rapidly  whirled  on 
a  vertical  axis.  The  force  of  adhesion  with  which  the  liquid  is  held 
between  the  particles  of  the  solid  or  within  its  pores  is  insufficient  to 
drag  the  liquid  round  in  a  circular  path.  The  result  is  that  the  liquid 
recedes  farther  and  farther  from  the  axis  until  it  finally  reaches  the 
outer  surface,  where  it  flies  out  through  the  openings  of  the  vessel. 
Centrifugal  machines,  acting  on  this  principle,  are  used  to  extract 
honey  from  the  comb,  to  separate  the  sirup  from  sugar  in  the  process 
of  refining,  to  dry  clothes  after  washing,  to  separate  cream  from  new 
milk,  etc.  The  separation  of  cream  depends  upon  the  fact  that  it 
is  less  dense  than  milk.  The  denser  milk  has  a  greater  centrifugal 
tendency,  and  consequently  moves  away  from  the  axis.  The  cream 
is  thus  crowded  toward  the  center,  where  it  is  drawn  off. 

124.  "  Centrifugal  Force."  -  When  a  person  who  is 
unacquainted  with  the  laws  of  dynamics  sees  a  body  vio- 
lently leave  a  curved  path  and  fly  off  at  a  tangent,  or  over- 
turn, he  naturally  assumes  that  this  behavior  is  due  to 
some  force  which  pulls  the  body  out  of  its  course.  Hence 
the  idea  that  centrifugal  motion  is  due  to  "  centrifugal 
force."  This  is  a  fundamental  error,  which  the  pupil 
who  has  mastered  the  preceding  work  of  this  chapter  will 
not  fail  to  detect.  The  error  consists  in  the  supposition 
that  a  body  moving  in  a  curved  path  tends  of  itself  to  con- 
tinue in  that  path,  and  that  it  will  do  so  unless  pulled  out 
of  it.  The  truth,  as  we  know,  is  precisely  the  contrary; 
namely,  that  a  moving  body  tends  of  itself  to  pursue  a 


140 


DYNAMICS 


straight  course,  and  will  do  so  unless  pulled  or  pushed  out 
of  it  into  a  curve.  It  follows  that  "  centrifugal  force/'  in 
the  above  sense,  is  pure  fiction,  and,  of  course,  does  not 
enter  into  any  correct  discussion  of  curvilinear  motion. 

Centrifugal  force  is  brought  from  the  realm  of  fiction 
to  that  of  fact  when  defined  as  the  reaction  to  the  cen- 


FIG.  121.  —  "Loop  the  Loop." 

tripe tal  force;  and  this  is  its  only  scientific  meaning.  The 
outward  thrust  of  a  car  on  the*  outer  rail  of  a  curve  is  the 
centrifugal  force  which  gives  rise  to  the  centripetal  reac- 
tion of  the  rail  on  the  car.  The  motion  of  the  car  is  deter- 
mined by  the  centripetal  thrust  on  it.  The  centrifugal 
thrust  is  indeed  very  real,  but  its  effect  is  expended  on  the 
rail  and  the  road-bed;  hence  the  necessity  for  a  strongly 
built  track. 

PROBLEMS 

1.  What  precautions  are  necessary  in  making  a  turn  on  a  bicycle  on  a 
slippery  pavement?     Discuss  these  precautions  as  illustrations  of  the  laws 
of  curvilinear  motion. 

2.  Must  a  light  and  a  heavy  bicycle  rider  lean  equally  or  unequally  in 
turning  the  same  curve  at  the  same  speed?     Give  reasons. 

3.  Are  tracks  inclined  more  or  less  on  sharp  curves  than  on  long  ones? 
What  law  is  illustrated? 


UNIVERSAL    GRAVITATION  141 

4.  Discuss  in  definite  and  accurate  terms  the  overturning  of  a  car, 
running  at  high  speed  on  a  curve. 

5.  What  is  the  percentage  of  increase  of  the  centrifugal  thrust  on  a  rail- 
road curve  when  the  velocity  of  a  train  is  increased  from  40  mi.  per  hour 
to  60  mi.  per  hour? 

6.  How  would  you  determine  experimentally  the  proper  inclination  of  a 
bicycle  race  track  on  a  curve? 

7.  Just  what  does  a  boy  do,  and  why,  to  change  his  direction  suddenly 
while  running? 

8.  Discuss  the  mechanics  of  "looping  the  loop"  (Fig.  121). 

9.  A  ball  weighing  2  kg.  is  suspended  from  a  cord  50  cm.  long,  and  made 
to  revolve  in  a  circle  whose  radius  is  30  cm.     (Fig.  119).     Compute  (a)  the 
centripetal  force  upon  the  ball,  and  (6)  the  tension  upon  the  cord. 

125.  Universal  Gravitation.  —  The  history  of  science 
tells  no  more  interesting  or  instructive  story  than  that  of 
Newton's  discovery  of  the  law  of  universal  gravitation  and 
the  work  of  his  predecessors,  which  made  that  discovery 
possible.*  Until  the  sixteenth  century  it  was  universally 
believed  that  the  earth  was  fixed  in  space,  and  that  the  sun, 
moon,  planets,  and  fixed  stars  revolved  in  the  heavens 
round  it.  This  view  was  overthrown  by  a  German  monk, 
named  Copernicus  (1473-1543),  who  taught  that  the  sun 
is  the  center  of  a  system  of  planets  which  revolve  round  it, 
and  that  the  earth  itself  is  one  of  them.  He  held,  however, 
to  the  old  view  that  the  orbits  of  the  planets  were  circles. 
The  next  advance  in  astronomical  science  was  due  to  Tycho 
Brahe  (1546-1601),  a  Danish  astronomer,  whose  observa- 
tions of  the  planetary  motions,  extending  over  a  period  of 
twenty  years,  were  much  more  extensive  and  accurate 
than  any  that  had  been  made  before.  These  observations 
were  subjected  to  a  searching  analysis  by  Johannes  Kep- 
ler (1571-1630),  "a  born  speculator  and  thinker,"  who 
"  after  more  than  four  years  of  assiduous  computation, 

*  This  story  is  admirably  told  in  Sir  Oliver  Lodge's  '  Pioneers  of  Science." 


142  DYNAMICS 

and  after  trying  more  than  nineteen  imaginary  paths  and 
rejecting  each  because  it  was  more  or  less  inconsistent  with 
observation,"  at  last  discovered  that  the  orbit  of  a  planet 
is  an  ellipse,  with  the  sun  at  one  of  its  foci  (Fig.  122).  This 
is  Kepler's  first  law  of  planetary  motion.  Further  years 
of  labor  brought  as  their  reward  the  discovery  of  his  sec- 
ond and  third  laws  of  planetary  motion. 

At  last  it  was  known  what  the  motion  of  a  planet  is;  but 
why  planets  move  thus,  rather  than  in  any  other  fashion, 
was  a  question  that  required  the  genius  of  Newton  to 
answer.  Previous  attempts  at  an  explanation  had  been 

based  upon  the  wrong  idea 
that  force  (unbalanced  force) 
is  necessary  to  maintain  a  body 
in  motion.  It  was  therefore 
supposed  that  a  force  of  some 
sort  must  act  on  a  planet  in 
its  line  of  motion  to  push  or 
pull  it  along.  Newton  knew 
FIG.  122. — Orbital  Motion  of  a  that  such  a  force .  is  unnec- 

essary;  but  Kepler  had  shown 

that  the  speed  of  a  planet  is  slightly  variable,  steadily 
increasing  as  the  planet  moves  from  the  farthest  point 
of  its  orbit,  B  (Fig.  122),  to  the  nearest  point,  A,  and  stead- 
ily decreasing  from  the  nearest  point  to  the  farthest  again, 
as  expressed  in  his  second  law  of  planetary  motion.  New- 
ton's problem,  therefore,  was  to  account  for  the  elliptical 
form  of  orbits  and  the  law  of  speed  in  them,  assuming  that 
the  general  laws  of  dynamics  hold  in  the  universe  at  large. 
He  proved  that  a  central  force,  directed  constantly  toward 
the  sun  and  varying  inversely  as  the  square  of  the  distance 
from  it,  would  accomplish  these  results,  and  that  a  force 
of  any  other  description  would  not.  He  proved  further 


UNIVERSAL   GRAVITATION  143 

that  the  moon  is  held  in  its  orbit  by  a  like  force,  directed 
toward  the  earth,  and  that  this  force  is  identical  with  the 
well  known  force  of  gravity,  which  makes  bodies  fall  and 
gives  them  their  weight.  The  solution  of  these  problems 
and  others  of  a  similar  character  showed  the  existence  of 
a  universal  attraction  or  gravitation,  as  expressed  in  the 
following  law:  Every  particle  of  matter  in  the  universe  at- 
tracts every  other  particle  with  a  force  whose  direction  is  that 
of  the  line  joining  them,  and  whose  magnitude  is  directly  pro- 
portional to  the  product  of  their  masses,  and  inversely  pro- 
portional to  the  square  of  the  distance  between  them. 

According  to  the  law,  there  is  a  gravitational  attraction 
between  all  bodies,  large  or  small.  Yet  even  between 
masses  of  several  hundred  pounds  it  is  exceedingly  small 
—  small  beyond  all  ordinary  means  of  detecting  it.  The 
greatest  ingenuity  and  experimental  skill  have  been  exer- 
cised by  various  scientists  during  the  past  century  in  accu- 
rately measuring  the  attraction  between  known  masses, 
varying  from  a  fraction  of  an  ounce  to  several  hundred 
pounds.  From  the  results  of  these  experiments  it  is  known 
that  two  spheres  of  cast  iron,  each  1.8  m.  in  diameter, 
would  attract  each  other  with  a  force  of  i  g.  when  placed 
close  together.  (Actual  contact  is  unnecessary.)  Such 
spheres  would  weigh  22,000  kg.  or  22  metric  tons  each. 

Concerning  the  cause  of  gravitation,  science  is  still  in  the  dark. 
It  acts  without  visible  or  material  connection  between  the  attract- 
ing bodies;  yet  we  must  suppose  that  there  is  something  pervading  all 
space,  by  means  of  which  and  through  which  it  is  exerted.  It  is 
inconceivable  that  two  bodies,  not  in  contact,  should  be  able  to  act 
upon  each  other  with  absolutely  nothing  between  them.  Since 
gravitation  acts  undiminished  in  a  vacuum,  and  beyond  the  limits 
of  the  atmosphere,  it  is  clear  that  the  means,  or  medium,  for  the 
transmission  of  gravitational  force  is  not  a  solid,  a  liquid,  or  a  gas, 
and  hence  is  not  matter  in  any  of  its  ordinary  forms. 


144  DYNAMICS 

126.  Applications  of  the  Law.  —  Center  of  Gravitation.  —  Newton 
proved  that  the  attraction  between  a  sphere  and  any  other  body  is 
the  same  as  it  would  be  if  the  entire  mass  of  the  sphere  were  con- 
centrated at  its  center.  The  attraction  between  two  spheres,  as  the 
earth  and  the  moon,  is,  therefore,  inversely  proportional  to  the  square 
of  the  distance  between  their  centers.  In  considering  the  earth's 
attraction  for  any  body  upon  its  surface,  the  distance  is  to  be  taken 
as  the  earth's  radius,  which,  in  round  numbers,  is  4000  mi. 

Gravitation  a  Mutual  Action.  —  Gravitation  between  any  two  bodies 
is  a  mutual  action,  in  agreement  with  the  third  law  of  motion.  A 
pound  mass  attracts  the  earth  with  a  force  of  one  pound.  The  earth 
and  the  moon,  by  their  mutual  attraction,  produce  equal  changes  of 
momentum  in  each  other;  but  the  mass  of  the  earth  is  80  times 
that  of  the  moon,  and  its  acceleration  is  consequently  sV  as  great. 
The  earth  and  the  moon,  in  fact,  revolve  in  the  same  direction  round 
their  common  center  of  gravity,  which,  as  it  divides  the  distance 
between  the  centers  of  the  two  bodies  inversely  as  their  masses,  lies 
within  the  mass  of  the  earth  about  noo  mi.  below  the  surface. 
(This  motion  of  the  earth  has  nothing  whatever  to  do  with  its  rota- 
tion on  its  axis.) 

The  sun's  attraction  deflects  the  earth  from  a  straight  course  by 
about  one  ninth  of  an  inch  in  a  second,  while  the  earth  is  traveling 
nearly  19  mi.  The  mass  of  the  earth  is  so  great  that  the  force 
required  to  produce  even  so  slight  a  change  of  direction  is  incon- 
ceivable, being  no  less  than  3,600,000  millions  of  millions  of  tons 
(36  with  seventeen  ciphers).  The  equal  pull  of  the  earth  upon  the 
sun  moves  it  less  than  an  inch  in  a  month,  the  mass  of  the  sun  being 
332,000  times  that  of  the  earth.  If  the  invisible  and  unknown  mech- 
anism of  gravitation  between  earth  and  sun  were  replaced  by  a 
cable  of  the  strongest  steel,  such  as  is  used  in  suspension  bridges,  that 
cable  would  have  to  be  3000  miles  in  diameter,  and  even  then  would 
be  strained  to  the  breaking  point. 

Rotation  of  the  Earth.  —  A  spinning  top  is  brought  to  rest  by  the 
resistance  of  the  air  and  the  friction  upon  the  peg.  The  earth  rotates 
on  its  axis  without  friction;  hence  its  rate  of  rotation  remains  con- 
stant without  the  action  of  any  force  to  maintain  it. 

If  the  earth  were  fluid  and  were  not  rotating,  the  gravitation  of 
its  particles  would  cause  it  to  assume  the  form  of  a  perfect  sphere. 


UNIVERSAL    GRAVITATION  145 

The  rotation  of  a  fluid  planet  would  cause  it  to  bulge  at  the  equator 
and  flatten  at  the  poles,  as  a  result  of  the  greater  centrifugal  tendency 
in  equatorial  regions,  where  the  velocity  of 
rotation  is  greatest.  This  is  illustrated  in 
Fig.  123,  which  represents  a  section  of  the 
earth  taken  through  the  axis  of  rotation  MN. 
(The  departure  from  the  spherical  shape  is 
greatly  exaggerated.)  The  earth  assumed  its 
present  form,  disregarding  minor  inequali- 
ties, 'while  still  fluid;  and,  as  a  result  of  its 
rotation,  the  polar  radius  is  nearly  13.5  mi. 
less  than  the  equatorial.  If  the  earth  were  FIG.  I23. 

to  stop  rotating,  the  Mississippi  River  would 

flow  toward  the  north,  for  its  mouth  is  farther  from  the  center  of 
the  earth  than  its  source  is. 

Variation  of  Weight.  —  All  bodies  on  the  earth  must  be  acted 
upon  by  a  centripetal  force  to  carry  them  round  with  the  earth  in 
its  rotation.  A  certain  portion  of  gravity  is  thus  employed,  and  only 
the  remainder  of  it  is  sensible  as  weight.  The  necessary  centripetal 
force  increases  from  zero  at  the  poles  to  ^-5  of  gravity  at  the 
equator.  Since  289  is  the  square  of  17,  and  centripetal  force  varies 
as  the  square  of  the  velocity,  it  follows  that,  if  the  earth  rotated  17 
times  as  fast  as  it  does,  bodies  at  the  equator  would  be  on  the  point 
of  flying  off  at  a  tangent,  and  would  weigh  nothing. 

There  is  a  further  cause  for  decrease  of  weight  as  a  body  is  taken 
toward  the  equator,  namely,  the  increasing  distance  from  the  earth's 
center.  For  this  reason  alone,  a  given  mass  weighs  ^-5  less  at  the 
equator  than  at  either  pole;  and,  in  consequence  of  rotation  and 
increase  of  distance  together,  it  weighs  about  jiv  less. 

PROBLEMS 

1.  (a)  Would  the  variation  of  weight  at  different  latitudes  be.  indicated 
by  any  form  of  balance  by  which  the  object  weighed  is  balanced  by 
"weights"?     (6)  Would  it  be  indicated  by  an  accurate  spring  balance? 
Give  reasons  for  each  answer. 

2.  What  fraction  of  its  weight  would  an  object  lose  when  taken  from 
sea-level  to  a  height  of  4  mi.? 

3.  How  does  it  follow  from  the  law  of  universal  gravitation  that  mass 
and  weight,  at  any  one  place  on  the  earth,  are  proportional? 


146  DYNAMICS 

4.  The  distance  of  the  moon  from  the  earth  is  240,000  mi.  (a)  How 
does  the  force  of  gravity  at  this  distance  compare  with  its  value  at  the 
surface  of  the  earth?  (6)  What  is  the  moon's  acceleration  toward  the  earth? 
(c)  How  far  is  the  moon  deflected  from  a  straight  course  in  one  second 
by  the  earth's  attraction? 

6.  Why  does  the  atmosphere  not  offer  resistance  to  the  rotation  or  to  the 
revolution  of  the  earth? 

6.  Is  the  acceleration  of  a  falling  body  due  to  the  whole  of  the  earth's 
attraction  or  to  the  part  that  we  call  weight? 

7.  What  would  be  the  subsequent  motion  of  the  moon  and  the  planets  if 
gravitation  should  suddenly  cease  to  act  upon  them? 

8.  The  average  specific  gravity  of  the  whole  earth  is  about  5.53.     (a) 
How  would  gravity  compare  with  its  present  value  if  the  average  density  of 
the  earth  were  equal  to  the  density  of  water?     (b)  What  would  be  the 
acceleration  of  a  falling  body  in  that  case? 

9.  The  diameter  of  Mars  is  4230  mi.  and  its  mass  is  approximately  one- 
ninth  of  the  earth's  mass.     How  does  gravity  upon  its  surface  compare  with 
gravity  upon  the  earth? 

127.  The  Pendulum.  —  Any  suspended  body  that  is 
free  to  swing  to  and  fro,  or  vibrate,  is  called  a  pendulum. 
A  pendulum  consisting  of  a  small  sphere  of  some  dense 
material,  suspended  from  a  fixed  support  by  a  slender 
thread  or  wire,  is  approximately  a  simple  pendulum.  Any 
pendulum  having  an  appreciable  portion  of  its  mass  else- 
where than  in  a  compact  mass  or  bob  at  the  end  is  a  com- 
pound pendulum.  Pendulums  for  other  than  experimental 
purposes  are  always  compound.  A  complete  swing  of 
a  pendulum  in  one  direction  is  called  a  vibration.  The 
period  of  a  pendulum  is  the  time  required  for  one  vibra- 
tion, and  is  measured  in  seconds.  The  amplitude  of  a 
pendulum  vibration  is  half  the  angle  or  half  the  arc  through 
which  it  swings.  The  length  of  a  simple  pendulum  is  (very 
approximately)  the  distance  from  the  point  of  suspension 
to  the  center  of  the  bob.  The  length  of  a  compound  pen- 
dulum is  denned  as  the  length  of  a  simple  pendulum  having 


THE   PENDULUM 


147 


the  same  period.  (It  will  be  found  by  trial  that  this  is 
greater  than  the'  distance  from  the  point  of  suspension 
to  the  center  of  gravity  of  the  o 

pendulum   and  is   less   than   its 
entire  length.) 

The  Motion  of  a  Pendulum. — After 
a  pendulum  has  been  drawn  aside  and 
released,  the  bob  is  under  the  action 
of  its  weight  and  the  tension  of  the 
thread  (friction  being  disregarded). 
As  the  pull  of  the  thread  acts  always 
at  right  angles  to  the  path  of  the  bob, 
its  only  effect  is  a  continuous  change 
of  direction.  The  weight  of  the  bob 
may  be  resolved  into  two  components, 
p  and  /  (Fig.  124),  at  any  point  of  the 
path.  The  component  p  causes  a  part  of  the  tension  on  the  thread, 
but  does  not  affect  the  motion  of  the  pendulum;  the  tangential 
component  /  accelerates  the  speed  as  the  bob  descends  and  retards 
it  as  the  bob  rises.  It  is  evident  that  the  tangential  force  decreases 
toward  the  lowest  point  of  the  path,  where  it  is  zero,  and  that  it  has 
equal  values  at  equal  distances  on  the  two  sides  of  this  point.  Hence, 
if  this  were  the  only  force  affecting  the  speed  of  the  bob,  it  would 
rise  exactly  as  far  as  it  descends,  and  its  vibration  would  continue 
indefinitely.  It  is  brought  to  rest  by  the  friction  of  the  air  and  the 
friction  at  the  point  of  support. 

128.  Laws  of  the  Pendulum.  —  The  laws  of  the  pendu- 
lum, as  determined  either  by  experiment  or  by  mathemat- 
ical analysis,  based  on  the  second  law  of  motion,  are  as 
follows : 

i.  The  period  of  a  pendulum  is  the  same  (to  an  exceed- 
ingly close  approximation)  for  all  amplitudes  less  than  4°;* 
for  larger  amplitudes,  the  period  increases  very  slightly 
with  increase  of  amplitude. 


148  DYNAMICS 

2.  The  period  of  a  pendulum  is  not  affected  by  its  mass 
or  the  kind  of  material  of  which  it  is  made. 

3.  The  period  of  a  pendulum  is  proportional   to   the 
square  root  of  its  length;  or,  the  square  of  the  period  is 
proportional  to  the  length.     Let  h  and  k  denote  the  lengths 
of  any  two  pendulums  and  t\  and  k  their  periods;  then 

ti'.  h::  \/  lim.  \/fe,  or  /i2:  t?\\  l\\  lz.  (17) 

4.  The  period  is  inversely  proportional  to  the  square 
root  of  the  acceleration  of  a  falling  body;  or 

til  k::  V  gz'.-V  gi-  (18) 

Mathematical  analysis  shows  that,  for  small  amplitudes 
of  vibration,  the  period  of  a  pendulum  is  given  by  the 
formula 

t  =  ' 

in  which  IT  denotes  the  ratio  of  the  circumference  of  a  circle 
to  its  diameter  (=  3.1416).  (Let  the  pupil  derive  the  pro- 
portions 17  and  18  from  this  formula.) 

The  effect  of  a  change  in  the  force  of  gravity  was  first 
recognized  when,  in  1671,  a  clock  was  taken  from  Paris 
to  French  Guiana,  on  the  northern  coast  of  South  America, 
for  use  in  astronomical  observations.  The  clock  lost  two 
and  a  half  minutes  daily  in  its  new  location.  The  pendu- 
lum was  shortened  to  correct  its  rate;  but  it  had  to  be 
lengthened  again  when  the  clock  was  taken  back  to  Paris. 
The  effect  of  an  increase  in  the  force  of  gravity  can  be  illus- 
trated experimentally  with  a  pendulum  having  an  iron 
bob.  By  holding  an  end  of  a  strong  bar  magnet  under  and 
near  the  bob,  its  motion  will  be  controlled  by  its  weight 
and  the  attraction  of  the  magnet,  acting  together,  the  lat- 
ter force  being  equivalent  to  an  increase  of  gravity.  With 


THE    PENDULUM 

a  small  amplitude  of  vibration,  the  bob 
does  not  swing  beyond  the  strong  attraction 
of  the  magnet,  and  the  period  is  consider- 
ably shortened. 

129.  Uses  of  the  Pendulum.  —  The  principal 
use  of  the  pendulum  is  to  regulate  the  motion  of 
clocks.  The  wheels  of  a  clock  are  driven  by  a 
weight  or  a  spring.  The  last  or  end  wheel  of  the 
train  is  trie  escapement  wheel  D  (Fig.  125),  the 
teeth  of  which  come  in  contact  with  the  projecting 
ends  of  a  curved  piece,  called  the  escapement. 
As  the  pendulum  swings,  it  rocks  the  escapement 
to  and  fro,  permitting  only  one  tooth  of  the  wheel 
to  pass  at  a  time.  Each  tooth,  as  it  passes,  exerts 
a  slight  impulse  on  the  escapement;  and  this  im- 
pulse, transmitted  to  the  pendulum,  maintains  its 
motion.  A  clock  is  regulated  by  means  of  a  thread 
and  nut  at  the  lower  end  of  the  pendulum.  The 
bob  is  raised  or  lowered  by  turning  this  nut. 

A  compound  pendulum  of  special  construction 
is  used  in  determining  the  acceleration  due  to 
gravity  at  different  places.  The  length  and  the 
period  of  the  pendulum  are  determined  very  accu- 
rately, and  their  values  substituted  in  the  pendulum  formula, 
which  the  value  of  g  is  then  computed. 

PROBLEMS 

1.  How  would  the  expansion  of  the  rod  of  a  pendulum  in  summer  and 
its  contraction  in  winter  affect  the  rate  of  a  clock  if  the  height  of  the  bob 
were  not  adjusted  to  compensate  the  expansion  and  contraction? 

2.  What  is  the  usual  shape  of  the  bob  of  a  clock  pendulum?    What  is 
the  advantage  of  this  shape? 

3.  What  is  the  length  of  a  pendulum  that  beats  seconds  (/  =  i)  at  a 
place  where  the  value  of  g  is  980  cm.  per  second  per  second  ? 

Suggestion.  —  Substitute  the  values  of  /  and  g  in  the  pendulum  formula, 
and  solve  for  /. 

4.  Account  for  the  fact  that  the  period  of  a  pendulum  is  independent 
of  its  mass. 


FIG.  125. 


from 


DYNAMICS 


6.   Referring  to  Fig.  126,  show  why  the  period  of  a  pendulum  is  in- 
creased  by  increasing  its  length. 

6.  Why  does  a  pendulum  of  given  length 
vibrate  more  rapidly  at  a  place  where  the  force 
of  gravity  is  greater? 

7.  Why  did   the  clock  which  was  taken 
from  Paris  to  Cayenne  (Art.  128)  lose  time? 

8.  Find  the  lengths  of  the  pendulums  whose 
periods  are  .7  sec.  and  1.5  sec.  respectively. 

9.  Find  the  periods    of  pendulums  whose 

lengths  are   20  cm.  and  250  cm.   re- 
spectively. 

10.  If  you  have  the  opportunity  to 
experiment  with  an  old  clock,  study  its 
mechanism  and  observe  its  behavior 
when  the  pendulum  bob  is  removed  and 
also  when  both  the  pendulum  and  the 
escapement  are  removed. 


FIG.  126. 


IV.    WORK   AND   KINETIC   ENERGY 

130.  Mechanical  Work  and  its  Effects.  —  When  a  man 
carries  a  hod  of  bricks  up  a  ladder,  he  does  a  certain  amount 
of  work.  This  amount  is  doubled  if  the  size  of  the  load 
is  doubled,  or  if  it  is  carried  to  twice  the  height.  If  both 
the  load  and  the  height  to  which  it  is  carried  are  doubled, 
the  amount  of  work  done  is  increased  fourfold.  Work, 
in  its  common  meaning  as  applied  to  physical  labor  and  to 
the  work  done  by  machines,  depends  upon  two  factors, 
namely,  the  amount  of  force  exerted  and  the  distance 
through  which  it  is  exerted.  It  is  therefore  measured  by 
the  product  of  these  factors.  If  the  body  acted  upon  does 
not  move,  no  work  is  done  upon  it.  A  hod-carrier  is  not 
working,  however  long  he  may  stand  with  a  load  of  bricks 
on  his  shoulder.  The  product  of  force  and  the  time  dur- 
ing which  it  acts  can  not  be  taken  as  the  measure  of  work. 
In  carrying  a  load  up  a  ladder  the  force  exerted  is  the  same 


WORK   AND    KINETIC    ENERGY  151 

whether  the  time  occupied  is  20  seconds  or  a  minute;  hence 
the  product  of  the  force  and  the  distance  measures  the  work 
done,  regardless  of  the  time. 

Daily  life  affords  innumerable  examples  of  mechanical 
work.  In  most  cases  this  work  consists  in  maintaining 
the  motion  of  bodies  against  friction  or  against  gravity, 
which  tends  to  stop  them.  A  horse  in  hauling  a  load  does 
work  against  friction  on  a  level  road,  and  against  both 
friction  and  gravity  in  going  up  grade.  In  going  down 
grade,  gravity  does  work  against  friction,  relieving  the  horse 
of  part  or  all  of  the  task. 

It  is  a  less  evident  fact  that  work  must  also  be  done  in 
imparting  and  in  destroying  motion;  for  this  work  is  com- 
monly insignificant  in  amount  compared  with  the  work 
done  in  maintaining  motion  over  long  distances,  as  in  haul- 
ing loads.  In  some  cases,  however,  the  amount  of  work 
done  in  starting  and  in  stopping  a  body  is  many  times 
as  great  as  the  work  done  in  the  interval  between,  e.g.  in 
throwing  and  catching  a  swift  ball.  The  work  done  upon 
the  ball  during  its  flight,  by  its  weight  and  the  resistance 
of  the  air,  is  practically  negligible. 

131.   Positive  and  Negative  Work.    Kinetic  Energy. — A 

force  acting  on  a  body  in  the  direction  of  its  motion  is  said 
to  do  positive  work  upon  the  bpdy;  if  acting  in  the  opposite 
direction,  it  does  negative  work.  Positive  work,  then, 
consists  in  maintaining  motion  against  friction  or  other 
opposing  force,  or  in  imparting  motion,  or  in  doing  both 
at  the  same  time.  Negative  work  consists  in  opposing 
and  in  reducing  the  motion  of  bodies.  Positive  work  in 
excess  of  negative  work  upon  a  body  increases  its  speed; 
negative  work  in  excess  of  positive  work  decreases  its 
speed;  equal  positive  and  negative  work  leaves  its  speed 


152  DYNAMICS 

unchanged.  Excess  of  positive  work  over  negative  work 
is  stored  up  in  the  moving  body,  to  be  paid  out  again  in  the 
act  of  stopping.  This  is  shown  by  the  fact  that  the  faster 
a  body  is  moving  the  farther  it  will  go  before  it  is  brought 
to  rest  by  friction  or  other  opposing  force. 

The  work  stored  in  a  moving  body  is  called  kinetic 
energy.  Kinetic  means  pertaining  to  motion;  and  the 
kinetic  energy  of  a  body  is  the  energy  it  possesses  by  vir- 
tue of  the  fact  that  it  is  a  moving  mass.  Energy  exists  in 
many  forms;  and  it  is  only  by  virtue  of  the  energy  it  pos- 
sesses in  one  form  or  another  that  any  body  can  do  work. 
The  energy  of  a  bent  bow  is  shown  by  its  ability  to  project 
an  arrow,  and  the  energy  of  a  coiled  spring  by  its  ability  to 
run  a  clock.  The  energy  of  the  wind  enables  it  to  turn 
windmills,  propel  ships,  uproot  trees,  etc.  The  energy 
of  coal,  wood,  and  oilis  utilized  by  the  steam-engine  in 
running  mills,  drawing  trains,  and  propelling  steamships. 
The  kinetic  energy  of  a  body  is  equal  to  the  excess  of  posi- 
tive work  that  has  been  done  upon  it  from  the  instant  of 
starting,  and  is  also  equal  to  the  work  that  the  body  is 
capable  of  doing  and  will  do  in  coming  to  rest. 

In  considering  the  effect  of  work  upon  one  body  only,  it  is  con- 
venient and  customary  to  say  that  the  work  is  done  by  a  force;  but 
we  know  that  a  second  body  is  always  necessary  to  exert  the  force, 
and,  strictly  speaking,  it  is  this  body  that  does  the  work.  "Doing 
work"  is,  in  /fact,  a  mutual  transaction,  like  buying  and  selling. 
When  a  marble  strikes  another,  the  first  loses  and  the  second  receives 
a  certain  store  of  kinetic  energy;  i.e.  the  positive  work  of  the  first 
marble  on  the  second  and  the  negative  work  of  the  second  on  the  first 
consists  in  the  transference  of  kinetic  energy  from  the  one  to  the  other. 

132.  A  Force  Perpendicular  to  the  Line  of  Motion  Does 
No  Work.  —  Thus  far  we  have  considered  only  the  work 
done  by  forces  acting  in  the  line  of  motion,  i.e.  in  the  direc- 
tion of  motion  or  in  the  opposite  direction.  A  force  acting 


WORK    AND    KINETIC    ENERGY  153 

at  right  angles  to  the  line  of  motion  does  no  work;  for 
mechanical  work  consists  in  increasing,  decreasing,  main- 
taining, or  opposing  the  motion  of  bodies,  and  such  a 
force  produces  none  of  these  results. 

It  is  true  that  weight  (a  vertical  force)  indirectly  opposes 
horizontal  motion  —  that  the  heavier  a  body  is  the  greater 
is  the  force  necessary  to  draw  it.  But  the"  opposing  force 
is  friction,  a  horizontal  force,  which  happens  to  be  propor- 
tional to  weight.  That  friction  is  no  part  of  weight  is 
further  evident  from  the  fact  that  it  can  be  almost  indefi- 
nitely reduced  by  means  of  ball  bearings,  an  even  track 
to  run  on,  etc. 

A  centripetal  force  in  curvilinear  motion  does  no  work, 
since  a  change  in  the  direction  of  motion  does  not  involve 
a  transfer  of  energy.  When  a  ball  is  revolved  on  a  table 
at  the  end  of  a  string,  the  inward  pull  can  be  supplied  at 
the  center  of  the  circle  without  moving  the  hand;  and  the 
hand  at  rest  does  no  work. 

133.   Oblique  Forces.     General  Rule  for  the  Measure  of 
Work.  —  To  find  the  work  done  by  a  force  whose  direction 
is  oblique  to  the  line  of  motion,  as  the 
weight  of  a  sphere  rolling  down  an  in- 
clined plane  (Fig.  127),  the  force  may  be 
resolved  into  two  components,  one  in  the 
line  of  motion,  /,  and  the  other  perpen- 
dicular to  it,  p.     Only  the  first  of  these 
components  does  work. 

Thus  if  d  denotes   the  length  of   the 

%J 

plane  AB,  the  work  done  upon  the  ball 

by  its  weight,  w,  while  it  is  rolling  down 

the  length  of  the  plane  is  fd.     If  h  denotes  the  height  of 

the  plane  J5C,  it  follows  from  similar  triangles  that/  :w  :: 


154  DYNAMICS 

h  :  d,  or  fd  =  wh.  Hence  the  work  done  is  also  measured 
by  wh,  i.e.  by  the  product  of  the  whole  force  and  the  dis- 
tance passed  over  in  its  own  direction. 

The  above  relations  are  general,  hence :  The  work  done  by 
an  oblique  force  is  measured  either  (i)  by  the  product  of  the 
component  force  in  the  line  of  motion  and  the  whole  distance 
passed  over  while  the  force  is  applied,  or  (2)  by  the  product  of 
the  whole  force  and  the  distance  through  which  it  is  applied 
in  its  own  direction.  The  choice  between  these  two  rules 
is  a  matter  of  convenience,  to  be  determined  by  the  nature 
of  the  problem.  When  the  force  and  the  actual  displace- 
ment of  the  body  are  in  the  same  or  in  opposite  directions, 
the  work  done  is  simply  the  product  of  the  two.  If  the 
force  is  variable,  the  average  force  exerted  through  the  dis- 
tance considered  must  be  taken.  In  general,  the  work  done 
by  a  force  /,  applied  through  a  distance  d  in  its  own  line  of 
direction,  is  given  by  the  formula, 

Work  =  fd.  (20) 

134.  Units  of  Work  and  Energy.  —  The  unit  of  work  is 
the  work  done  by  unit  force  while  its  point  of  application 
moves  unit  distance  in  the  line  of  the  force.  If  the  unit 
force  is  the  pound  and  the  unit  distance  the  foot,  the  unit 
of  work  is  called  the  foot-pound  (ft.-lb.).  Similarly  we 
have  the  gram-centimeter  (g.-cm.)  and  the  kilogram-meter 
(kg.-m.)  as  the  gravitational  units  of  work  in  the  C.  G.  S. 
system.  The  foot-pound  is  the  unit  of  work  employed  by 
American  engineers.  The  absolute  units  of  work  are  the 
foot-poundal  and  the  dyne-centimeter.  (Define  each  of 
these  units.)  The  dyne-centimeter  is  called  an  erg  (from 
the  Greek  ergon,  work).  It  is  extremely  small,  and  in 
actual  practice  a  larger  unit,  equal  to  10,000,000  ergs,  is 


WORK    AND    KINETIC    ENERGY  155 

used.     This  larger  unit  is  called  a  joule,  in  honor  of  the 
English  physicist,  James  Prescott  Joule. 

The  units  of  work  are  also  the  units  of  kinetic  energy. 

EXAMPLES.  —  i.  When  a  book  weighing  i  Ib.  is  lifted  4  ft.,  the 
work  done  against  gravity  is  4  ft.-lb.  This  is  equal  to  128.64  foot- 
poundals,  -553°3  kg.-m.,  55,303  g.-cm.,  54,197,000  ergs,  or  5.419? 
joules. 

2.  If  the  hammer  of  a  pile-driver  weighing  1000  Ib.  falls  25  ft. 
and  strikes  a  pile,  its  weight  does  25,000  ft.-lb.  of  work  upon  it  during 
its  fall.     Since  the  resistance  of  the  air  is  negligible,  this  work  is  all 
stored  in  the  hammer  as  kinetic  energy,  and  the  hammer  does  25,000 
ft.-lb.  of  work  upon  the  pile. 

3.  A  ball  weighing  4  oz.  is  thrown  vertically  upward  by  a  force 
of  10  Ib.  acting  through  a  distance  of  3  ft.     How  high  does  it  rise? 
The  projecting  force  does  10  X  3  or  30  ft.-lb.  of  positive  work  upon 
the  ball.     Hence  it  will  continue  to  rise  until  the  negative  work  of 
its  weight  amounts  to  30  ft.-lb.;  and,  since  its  weight  is  .25  Ib.,  the 
height  is  120  ft.  (=  30  -i-  .25)  from  the  point  of  starting,  or  117  ft. 
from  the  point  at  which  the  ball  is  released.     The  unbalanced  upward 
force  exerted  in  throwing  the  ball  is  9.75  Ib.,  and  this  imparts  9.75  X  3 
or  29.25  ft.-lb.  of  kinetic  energy.     This  kinetic  energy  is  lost  at  the 
rate  of  .25  ft.-lb.  for  every  foot  of  ascent,  and  enables  the  ball  to  rise 
117  ft.  above  the  point  of  release  before  coming  to  rest. 

4.  A  horse  pulls  with  a  force  of  190  Ib.  through  a  distance  of  10  ft. 
in  starting  a  load,  and  the  resistance  of  friction  is  130  Ib.  (a)  How 
much  work  is  done  by  the  horse?     (b)  What  does  this  work  accom- 
plish?    (c)  In  what  distance  will  friction  alone  stop  the  cart,  if  the 
horse  ceases  to  pull  at  the  end  of  the  ten  feet?  - 

(a)  Work   done   by   the   horse  =  190  X  10  =  1900  ft.-lb. 

(b)  Work  done  against  friction  =  130  X  10  =  1300  ft.-lb. 
Work  done  by  the  unbalanced  pull   =  kinetic  energy  imparted 

=  60  X  10  =  600  ft.-lb. 

(c)  Since  friction  in  stopping  the  cart  must  do  600  ft.-lb.  of 
negative  work,  the  distance  =  600  +  130  =  4.61  ft. 

135.  Measure  of  Kinetic  Energy.  —  The  kinetic  energy 
of  a  body  is  equal  to  the  positive  work  that  has  been  done 


156  DYNAMICS 

upon  it,  in  excess  of  the  negative  work,  from  the  instant 
of  starting,  and  is  also  equal  to  the  work  that  the  body  is 
capable  of  doing  in  coming  to  rest.  Hence  if  we  know 
either  of  these  quantities  of  work,  we  know  the  kinetic 
energy  of  the  body;  but,  since  kinetic  energy  consists  in 
the  motion  of  mass,  it  can  always  be  expressed  in  terms  of 
mass  and  velocity,  and  is  regularly  thus  expressed.  The 
formula  for  kinetic  energy  (K.  E.)  can  be  derived  by  con- 
sidering either  of  the  above-named  quantities  of  work  to 
which  it  is  equal. 

Let  /  denote  the  unbalanced  or  accelerating  force  in  the 
direction  of  motion,  acting  upon  a  mass  m,  and  a  the  accel- 
eration which  it  produces.  The  work  done  by  such  a 
force  consists  in  imparting  K.  E.;  hence  the  K.  E.  imparted 
is  measured  byfd. 

But  /  =  ma,  (Equation  13). 

i)2 
and  d  =  — ;  (Equation  6) 

from  which        fd  =  ma  X  —  =  \  mv2. 

Hence  K.  E.  =  \  mv2  (ergs  or  foot-poundals)  (21) 

Since  the  formula  /  =  ma  holds  only  for  absolute  units  of 
force  (the  dyne  or  the  poundal),  the  above  formula  gives 
K.  E.  in  ergs  or  foot-poundals,  according  as  mass  and 
velocity  are  expressed  in  English  or  metric  units.  With 

ma 

any  gravitational  unit  of  force,  /  =  — ;  hence 

I 

K.  E.  =  -  -  (g.-cm.,  kg.-m.,  or  ft.-lb.)  (22) 

In  deriving  the  formula  for  K.  E.  by  considering  the  amount  of 
work  that  a  moving  body  will  do  in  coming  to  rest,  the  procedure  is 


WORK    AND    KINETIC    ENERGY  157 

the  same  as  above ;  but  in  this  case  /  is  the  force  exerted  by  the  body 
through  the  distance  d  in  coming  to  rest,  and  the  acceleration  a  is 
negative. 

136.  Power.  —  The  rate  at  which  an  engine  or  other 
source  of  energy  is  capable  of  doing  work  is  called  its 
power.  Power  can  be  measured  in  foot-pounds  per  sec- 
ond, foot-pounds  per  minute,  kilogram-meters  per  second, 
etc.  The  customary  unit  of  power  is  the  horse-power 
(H.  P.),  which  is  equal  to  550  ft.-lb.  of  work  per  second, 
or  33,000  ft.-lb.  per  minute,  or  76  kg.-m.  per  second.  A 
twelve-horse-power  engine,  working  at  three  fourths  of 
its  full  capacity,  does  work  at  the  rate  of  9  H.  P.,  or  4950 
ft.-lb.  per  second. 

The  horse-power  was  denned  by  James  Watt,  the  in- 
ventor of  the  steam-engine,  and  was  his  estimate  of  the 
rate  at  which  a  draft-horse  is  capable  of  working.  The 
power  of  an  average  horse,  for  steady  work,  is  about 
.8  H.  P. 

PROBLEMS 

1.  What  is  the  relation  between  the  K.  E.  of  a  body  and  its  mass? 
Between  its  K.  E.  and  its  velocity? 

2.  Two  bodies  have  equal  kinetic  energy,  but  the  velocity  of  the  second 
is  three  times  that  of  the  first.     How  do  their  masses  compare? 

3.  A  body  is  thrown  vertically  upward,     (a)  What  fraction  of  its  initial 
kinetic  energy  remains  after  it  has  risen  to  one  half  the  height  to  which  it 
will  ascend?     (b)  What  fraction  of  its  initial  velocity  remains?     (c)  What 
fraction  of  the  initial  kinetic  energy  and  of  the  initial  velocity  remain  after 
the  body  has  risen  to  three  fourths  of  the  total  height? 

4.  (a)  A  boy  starting  at  rest  coasts  on  a  bicycle  down  a  hill  and  up 
another.     If  there  were  no  friction,  how  far  would  he  ascend  the  second  hill 
without  pedaling?     (6)  How  would  the  result  be  affected  if  either  hill  were 
steeper  than  the  other? 

6.  A  sphere  weighing  20  Ib.  is  set  rolling  on  a  level  surface  with  a  veloc- 
ity of  24  ft.  per  second.  The  resistance  to  motion  (friction)  is  i  Ib.  Find 
(a)  the  K.  E.  of  the  sphere  at  the  start;  (b)  the  loss  of  K.  E.  in  rolling  40 
ft.;  (c)  the  distance  the  ball  will  roll  before  stopping. 


158  DYNAMICS 

6.  A  projectile  weighing  500  Ib.  is  fired  from  a  cannon  with  a  velocity 
of  3000  ft.  per  second,  (a)  What  is  its  kinetic  energy?  (b)  What  was  the 
average  force  acting  upon  the  projectile  in  firing  it,  if  it  moved  a 

distance  of  18  ft.  before  reaching  the   mouth    of   the 

cannon? 

7.  In  whirling  a  body  round  the  hand  at  the  end 
of  a  string,  the  hand  is  moved  in  a  smaller  circle  in 
advance  of  the  body  whirled  (Fig.  128).  Show  how  this 
imparts  kinetic  energy  to  the  body. 

FlG  I2g  8.  A  body  weighing  100  Ib.  slides  30  ft.  down  a 

plane  inclined  at  an  angle  of  45°.  Friction  is  30  Ib. 
Find  (a)  the  accelerating  force;  (b)  the  K.  E.  at  the  end  of  the  thirty- 
foot  slide;  (c)  the  work  done  against  friction  in  that  distance. 

9.  (a)  Is  a  person  doing  work  against  gravity  in  carrying  a  load  over 
level  ground?  (b)  What  is  the  measure  of  the  work  done  when  the  load  is 
carried  uphill? 

10.  A  stone  weighing  1.5  Ib.  is  thrown  vertically  upward  by  means  of  a 
force  of  20  Ib.  acting  through  a  distance  of  3  ft.     How  high  will  it  rise  above 
the  starting-point? 

11.  A  body  weighing  15  kg.  falls  vertically  a  distance  of  20  m.     What  is 
its  kinetic  energy? 

12.  A  bullet  weighing  5  g.  and  having  a  velocity  of  300  m.  per  sec- 
ond strikes  a  log  and  penetrates  it  a  distance  of  10  cm.     What  average 
resistance  did  the  bullet  encounter  in  penetrating  the  log? 

13.  (a)  How  much  work  is  done  in  filling  a  reservoir  that  has  a  capacity 
of  looo  cu.  m.  if  the  water  must  be  raised  12  m.  to  discharge  it  into  the 
reservoir?     (b)  How  long  would  it  take  a  twelve-horse-power  engine  to  fill 
the  reservoir? 

14.  What  is  the  power  of  an  engine  that  is  capable  of  drawing  a  train  at 
the  rate  of  30  mi.  per  hour  against  a  resistance  of  6250  Ib.? 

15.  If  a  man  pumps  250  Ib.  of  water  per  minute  to  a  height  of  15  ft., 
(a)  how  many  foot-pounds  of  work  does  he  do  in  an  hour  ?      (b)  At  what 
rate  in  horse-power  is  he  working? 

16.  Distinguish  between  momentum  and  K.  E.     Is  the  force  that  is 
required  to  stop  a  moving  body  proportional  to  its  momentum  or  to  its  K.  E., 
if  the  body  is  brought  to  rest  in  a  given  time?    If  it  is  brought  to  rest  within 
a  given  distance? 

17.  The  standard  i2-in.  gun  of  the  United  States  Navy  fires  a  projectile 
of  icoo  Ib.  weight  with  a  velocity  of  2550  ft.  per  second.    The  new  i4-in.  gun 


MACHINES 

fires  a  i6oo-lb.  projectile  with  a  muzzle  velocity  of  2150  ft.  per  second. 
Find  the  muzzle  energy  of  each  projectile. 

18.  At  what  rate  in  ft.-lb.  per  second  is  a  draft-horse  working  when 
exerting  a  pull  of  175  Ib.  in  hauling  a  load  at  the  rate  of  3  mi.  per  hour? 

19.  Find  by  trial  the  rate  in  H.  P.  at  which  you  can  work  against  your 
own  weight  in  going  upstairs  (a)  as  you  usually  do,  (b)  when  running  up  as 
fast  as  you  can. 

20.  Find  the  K.  E.  of  the  ball  and  the  K.  E.  of  the  rifle  of  problem  27, 
p.  134.    Account  for  the  fact  that  the  K.E.  of  the  rifle  is  much  less  than  that 
of  the  ball,  while  their  momenta  are  equal. 

V.  MACHINES 

137.  General  Uses  of  Machines.  —  Any  contrivance  used 
in  doing  work,  from  the  simplest  hand  tool  to  the  printing- 
press  or  the  locomotive  engine,  is  a  ma- 
chine. In  general  terms,  the  purpose  of 
a  machine  is  either  to  secure  some  ad- 
vantage by  its  use  when  energy  is  to  be 
transferred  from  one  body  to  another,  or 
to  transform  energy  from  one  kind  into 
another.  For  example,  a  horse  can  raise 
a  load  vertically, 
with  the  aid  of  a 
rope  and  two  pul- 
leys to  transmit  the 
pull  and  change  its 
direction  (Fig.  129); 
and  a  crowbar  en- 
ables a  man  to  move 
a  heavy  object  with 
comparatively  little 
effort,  the  gain  of  force  being  attended  by  a  corresponding 
loss  of  speed  and  distance  (Fig.  67).  A  dynamo  receives 
mechanical  energy  from  an  engine  or  a  water-wheel  and 


FIG.  129.  —  Use   of   Fixed 
Pulleys. 


160  DYNAMICS 

transforms  it  into  electrical  energy,  for  convenient  trans- 
mission over  a  wire  to  some  distant  point,  where  it  may 
again  be  transformed  into  mechanical  energy  by  means  of 
a  motor,  and  utilized  for  running  a  street  car  or  for  other 
work. 

The  principal  advantages  to  be  derived  from  the  use  of 
machines  are  the  following: 

1.  They  permit  a  gain  of  force  at  the  expense  of  speed 
and  distance.     All  machines  for  exerting  great  forces  are 
constructed   on  this  principle.     Examples:   The   crowbar 
(Fig.  67),  rope  and  pulleys  (Fig.  142),  the  wedge  (Fig.  148), 
the  lifting  jack  (Fig.    150),  the  hydraulic  press  (Fig.  17), 
cranes,  derricks  (Fig.  141),  etc. 

2.  They  permit  a  gain  of  speed  and  distance,  with  a 
corresponding  loss  of  force.     Examples:  The  bicycle,  the 
sewing-machine,  lathes,   saws,  centrifugal   machines,  etc. 

3.  They  permit  any  desired  change  in  the  direction  of 
the  applied  force.     Examples:  Rope  and  pulley  (Fig.  129), 
beveled  gear-wheels  (Fig.  159). 

4.  They   enable    man   to    utilize    the    various   natural 
sources  of  energy.     For  example,  the  energy  of  winds  by 
windmills,   the   energy  of  water  by  water-wheels   (Figs. 
156-160),  the  energy  of  fuel  by  means  of  steam,  gas,  and 
other  engines  (Figs.  202-211). 

138.  Useful  and  Wasted  Work  of  Machines.  — A  machine 
can  transfer  or  transform  energy  only  as  it  receives  energy 
from  some  other  body;  it  is  never  itself  the  original  source 
of  work  or  energy.  An  engine  can  run  the  machinery  of  a 
factory;  but  steam  is  necessary  to  run  the  engine,  and  coal 
or  other  fuel  is  necessary  to  generate  the  steam.  An  ideally 
perfect  machine  would  do  work  without  loss  or  waste  within 


MACHINES  161 

itself;  and,  as  will  be  shown  later,  the  work  done  by  it 
would  be  the  full  equivalent  of  the  work  done  upon  it  — 
neither  more  nor  less.  This  is  a  general  principle  of  the 
greatest  scientific  and  practical  importance. 

No  machine  is  perfect,  in  the  above  sense.  The  motion 
of  the  parts  of  a  machine  always  develops  friction;  and 
friction  results  in  loss  of  energy,  or  wasted  work,  within 
the  machine  itself.  Further  losses  result  from  the  bend- 
ing and  vibration  of  the  parts  of  a  machine,  and  from  the 
stiffness  of  ropes  and  belts.  The  useful  work  of  a  machine 
is  thus  the  difference  between  the  work  done  upon  it 
and  the  work  lost  or  wasted  in  it. 

139.  Mechanical  Advantage  and  Efficiency  of  Machines. 
-The  source  from  which  a  machine  receives  energy  is 
called  the  agent.  In  hand  labor  the  person  who  operates 
the  machine  is  the  agent.  Steam  is  the  agent  that  supplies 
energy  to  the  steam-engine.  .Strictly  speaking,  it  is  the 
agent,  not  the  machine,  that  does  the  work.  The  body 
moved  by  a  machine  is  called  the  load.  The  opposing 
force  exerted  by  the  load  while  it  is  being  moved  is  called 
the  resistance.  If  the  work  consists  in  lifting  the  load, 
the  resistance  is  equal  to  its  weight.  The  force  exerted 
by  the  agent  upon  the  machine,  in  moving  the  load  with- 
out acceleration,  is  called  the  working  effort.  The  force 
that  would  be  necessary  if  the  machine  were  perfect  is 
called  the  static  effort,  for  it  is  just  the  force  that  would 
be  necessary  to  maintain  equilibrium  against  the  resist- 
ance. With  a  perfect  machine,  the  static  effort  and  the 
working  effort  would-be  the  same;  with  actual  machines 
the  working  effort  is  necessarily  the  greater. 

The  ratio  of  the  resistance  to  the  effort  is  called  the 
mechanical  advantage  of  a  machine.  If  a  crowbar  is  so 


162  DYNAMICS 

adjusted  that  it  overcomes  a  resistance  of  450  Ib.  when 
the  effort  applied  is  75  Ib.,  its  mechanical  advantage  for 
that  adjustment  is  6.  In  studying  mechanical  principles 
no  account  is  taken  of  the  imperfections  of  actual  machines, 
and  the  mechanical  advantage  is  consequently  taken  as 
the  ratio  of  the  resistance  to  the  static  effort.  In  actual 
practice  it  is  the  ratio  of  the  resistance  to  the  working 
effort.  The  mechanical  advantage  of  a  machine  is  de- 
termined by  the  relative  size  or  number  of  certain  parts 
of  it,  as  we  shall  see. 

The  ratio  of  the  useful  work  done  by  a  machine  to  the 
work  done  on  it  is  called  the  efficiency  of  the  machine. 
It  is  commonly  expressed  as  a  percentage.  The  efficiency 
of  a  perfect  machine  would  be  100%.  The  efficiency 
of  actual  machines  lies  in  most  cases  between  30%  and 
95%.  Its  determination  in  any  given  case  is  an  experi- 
mental problem. 

Let  Es  denote  the  static  effort  and  Ew  the  working  effort 
when  the  resistance  offered  by  the  load  in  uniform  motion 
is  R,  and  De  the  distance  through  which  the  effort  acts 
(the  effort  distance)  in  moving  the  load  through  the  dis- 
tance Dr  (the  resistance  distance).  Then  the  work  done 
by  the  agent  upon  the  machine  is  EwDe  (the  product  of 
the  working  effort  and  the  effort  distance);  and  the  work 
done  by  the  machine  upon  the  load  is  RDr  (the  product 
of  the  resistance  and  the  resistance  distance).  Then,  also, 

r> 

Mechanical  advantage  =  -IT.  (23) 

**j 

RDr 

Efficiency  =  —  —  -  (24) 


EwDe  =  RD,  +  wasted  work.  (25) 


MACHINES  163 

140.  The  Simple  Machines.  —  All  machines,  however 
complicated,  are  but  modifications  and  combinations  of 
one  or  more  of  the  six  simple  machines.    These  are  the  lever, 
the  wheel  and  axle,  the  pulley,  the  inclined  plane,  the 
wedge,  and  the  screw.     An  understanding  of  the  simple 
machines  is,   of  necessity,   the   first  step  in   mechanical 
knowledge. 

Each  of   the   simple   machines  presents   the   following 
problems: 

1.  To  find  its  mechanical  advantage  in  terms  of  cer- 
tain dimensions  of  it. 

2.  To  show  that,  as  an  ideal  machine,  it  transmits  energy 
without  gain  or  loss,  i.e.  to  show  that  EsDe  =  RDr.     This 
is  the  general  law  of  machines. 

3.  To  become  acquainted  with  some  of  its  common 
forms   and   their   uses. 

4.  To  determine  the  efficiency  of  the  machine  provided 
for  laboratory  study. 

141.  The  Lever.  —  A  bar  or  rod  which  turns  about  a 
fixed  support  or  axis,  in  transmitting  motion  from  one 
body  to  another,  is  called  a  lever,  and  the  support  is  called 
the  fulcrum.    For  convenience  in  discussing  their  use  levers 
are  grouped  into  three  classes.     In  levers  of  the  first  class 
(Fig.  131)  the  fulcrum  F  is  between  the  points  of  application 
of  the  effort  E  and  the  resistance  R;  in  levers  of  the  second 
class  (Fig.  132)  the  resistance  is  applied  between  the  fulcrum 
and  the  effort;  in  levers  of  the  third  class  (Fig.  133)  the 
effort  is  applied  between  the  fulcrum  and  the  resistance. 
Taking  the  classes  in  order,  the  fulcrum,  the  resistance, 
and  the  effort  are  respectively  between  the  other  two,  — 
a  fact  easily  remembered  from  the  initial  letters  FRE   (as 
in  the  word  free) . 


1 64  DYNAMICS 

The  mechanical  principle  of  the  lever,  of  whatever  form, 
is  that  of  moments  of  force  in  equilibrium  (Art.  78).  For 
the  effort  and  the  resistance  always  act  in  opposite  direc- 
tions round  the  fulcrum;  and,  disregarding  friction,  their 
moments  are  equal,  whether  the  lever  is  at  rest  or  in 
uniform  motion;  that  is, 

EAe  =  RAr,  or  R:  E::  Ae:  Ar,  (26) 

Ae  being  the  arm  of  the  effort  and  Ar  the  arm  of  the  resist- 
ance. Hence  the  mechanical  advantage  of  the  lever  is  deter- 
mined and  measured  by  the  ratio  of  the  arm  of  the  effort  to 
the  arm  of  the  resistance.  This  ratio  is  commonly  known 
as  the  leverage.  The  leverage  can  evidently  be  made  as 
great  or  as  small  as  is  desired,  by  simply  changing  the 
position  of  the  fulcrum  or  the  points  of  application  of  the 
effort  and  the  resistance. 

Applications  of  the  Lever. — The  lever  in  different  forms  is 
adapted  to  various  special  uses.  In  most  cases  it  enables  a 
given  force  to  overcome  a  resistance  several  times  as  great 
as  itself.  This  advantage,  which  is  known  as  a  gain  of 


A*- 
*-**— *-- — -A.- — -4 


_^_ __^ 


R 


FIG.  131.  —  Levers  of  the  First  Class.       FIG.  132.  —  Levers  of  the  Second  Class. 

force j  is  afforded  by  forceps,  pincers,  wire  cutters  (Fig.  1316) 
and  nutcrackers  (Fig.  1326);  all  of  which  are  double  levers 
having  the  arm  of  the  effort  longer  than  the  arm  of  the 
resistance.  A  lever  may  be  curved  or  angular,  as  a  claw- 


MACHINES  165 

hammer  when  used  in  drawing  a  nail  (Fig.  68).     Many 
forms  of  the  lever  are  designed  with  reference  solely  to 


*" — -A: * 

FIG  133.  —  Levers  of  the  Third  Class. 

convenience  or  adaptability  to  the  work  to  be  done,  the 
relative  value  of  the  effort  and  resistance  being  unimpor- 
tant. Tweezers,  coal  tongs,  sugar  tongs,  and  scissors  are 
examples. 

In  certain  applications  of  the  lever  the  advantage 
secured  is  a  gain  of  speed  and  distance,  i.e.  the  point  of 
application  of  the  resistance  moves  faster  and  farther  than 
the  point  of  application  of  the  effort.  This  is  well  exempli- 
fied by  the  movements  of  our  bodies  and  the  bodies  of 
animals  in  general.  The  movable  parts  of  the  skeleton 
are  levers;  the  joints  are  the  fulcrums.  The  muscles  are 
attached  to  the  bones  near  the  joints  by  means  of  tendons. 
A  muscle  acts  by  contracting  or  shortening.  This  causes 
the  bone  to  which  it  is  attached  to  move,  and  the  farther 
extremity  of  the  bone  moves  much  faster  and  farther  than 
the  point  to  which  the  muscle  is  attached  (Fig.  1336). 
Levers  of  this  type  are  used  in  such  instruments  as  the 
aneroid  barometer  to  magnify  small  motions. 

142.  The  Law  of  Work  for  an  Ideal  Lever.  —  Suppose  a 
load  R  (Fig.  134)  to  be  raised  through  a  vertical  distance 
Dr,  by  means  of  a  perfect  lever,  while  the  effort  E  acts 


i66 


DYNAMICS 


FIG.  134. — The  Law  of  Work. 


vertically  through  a 
distance  De.  (The 
arms  of  the  resistance 
and  effort  constantly 
change  during  the 
motion;  but  their 
ratio,  Ae:  Ar)  remains 
constant.  Why?) 
Since  the  lever  is 

assumed  to  be  perfect,  the  working  effort  is  equal  to  the 

static  effort  and 

R:E::Ae:Ar, 

while  the  work  is  in  progress. 

By  geometry,  De:  Dr::  Ae:  A,.  (Why?) 

From  the  two  proportions, 

R:  E::  De:  Dr  or  EDe  =  RDr. 

But  EDe  is  the  work  done  by  the  effort  and  RDr  is  the 
work  done  upon  the  load.  Hence  we  have  proved  that, 
in  doing  this  piece  of  work,  a  perfect  lever  would  transmit 
energy  without  gain  or  loss.  It  can  be  shown  that  this  is 
always  true  of  a  perfect  lever,  whatever  the  character  of 
the  work.  It  follows  that,  under  the  most  favorable  con- 
ditions, whenever  force  is  gained  by  means  of  a  lever  there 
is  a  proportionate  loss  of  speed  and  distance,  and  whenever 
speed  and  distance  are  gained  there  is  a  proportionate  loss 
of  force.  (Show  this.)  Evidently  a  lever,  at  least,  can 
render  no  assistance  as  a  part  of  a  "  perpetual-motion 
machine." 

PROBLEMS 

1.  In  using  scissors  is  greater  force  required  when  the  cutting  is  done  near 
the  tips  of  the  blades  or  near  the  handles?    Why? 

2.  Classify  the  following  levers,  and  state  in  each  case  whether  the  effort 


MACHINES 


167 


is  greater  or  less  than  the  resistance:  the   wheelbarrow,  oar,  fishing-rod, 
equal-arm  balance,  steelyard,  nutcracker. 

3.  Use  a  pencil  as  a  lever  of  the  first  class  to  move  a  book;  also  as  a 
lever  of  the  second  class. 

4.  In  which  class  or  classes  of  levers 
is  the  effort  necessarily  less  than  the 
resistance?  In  which  may  it  be  either 
greater  or  less? 


FIG.  135. — The  Foot  as  a  Lever. 


FIG.  136. 


5.  (a)  If  a  stone  offers  a  resistance  of  850  lb.,  what  leverage  will  be 
required  to  move  it  by  means  of  a  force  of  125  lb.?     (b)  If  the  stone  is 
moved  by  a  crowbar  5  ft.  long  used  as  a  lever  of  the  first  class,  the  effort  and 
the  resistance  being  applied  at  the  ends,  where  is  the  fulcrum? 

6.  A  person  rises  on  his  toes  by  the  action  of  the  calf  muscles,  which  pull 
on  a  tendon  attached  to  the  heel  bone  (Fig.  135).     To  what  class  of  levers 
does  the  foot  belong  when  thus  used?     What  relations  hold  between  the 
three  forces  and  the  two  distances?     Which  of  the  forces  is  equal  to  the 
person's  weight?     It  may  be  of  assistance  to  note  that  the  three  forces  are 
parallel  and  in  equilibrium.     This  action  of  the  foot  can  be  studied  experi- 
mentally by  means  of  two  short  boards,  AC  (Fig.  136),  and  a  stout  cord. 
The  force  exerted  in  thus  lifting  one's  self  can  be  measured  with  a  balance, 
as  shown  in  the  figure. 

143.  The  Wheel  and  Axle.  —  The  wheel  and  axle  (Fig. 
137)  consists  of  a  wheel  and  an  axle  or  cylinder,  turning 
as  one  body  on  the  same  axis.  In  the  modified  form  known 
as  a  windlass  (Fig.  138)  the  wheel  is  replaced  by  a  crank, 
which  serves  the  same  purpose.  The  figures  clearly  indi- 
cate the  use  of  this  machine  as  an  aid  in  manual  labor. 


i68 


DYNAMICS 


The  effort  is  applied  at  any  convenient  point  on  the  cir- 
cumference of  the  wheel  or  at  the  crank  handle,  and  the 
resistance  acts  at  the  circumference  of  the  axle. 


FIG.  137.  —  Wheel  and  Axle. 


FIG.  138.— Windlass. 


Mechanical  Advantage.  —  The  wheel  and  axle  may  be 
regarded  as  a  continuously  acting  lever  (Fig.  139),  the  radius 
of  the  wheel  being  the  arm  of  the  effort,  and  the  radius 
of  the  axle  the  arm  of  the  resistance.  If  the  machine  were 
perfect,  the  moments  of  the  effort  and  the  resistance  would 
be  equal,  both  for  equilibrium  and  for  uniform  motion; 
i.e.  we  should  have 

E  X  radius  of  wheel  =  R  X  radius  of  axle, 
or  R:  E::  radius  of  wheel :  radius  of  axle.  (27) 

The  mechanical  advantage  of  the 
wheel  and  axle  is,  therefore,  the 
ratio  of  the  radius  of  the  wheel  to 
the  radius  of  the  axle. 

The  Law  of  Work.— During  one 
complete  revolution  of  the  wheel 
the  effort  acts  through  a  distance 
equal  to  its  circumference,  and  the 
load  is  moved  a  distance  equal  to 


FIG.  140.  —  The  Capstan. 


MACHINES  169 

the  circumference  of  the  axle.  The  ratio  of  the  effort 
distance,  De,  to  the  resistance  distance,  Dr,  is  evidently 
the  same  for  any  number  of  revolution,  and  is  given  either 
by  the  ratio  of  the  circumferences  of  the  wheel  and  the  axle 
or  by  the  ratio  of  their  radii;  that  is, 

De:  Dr::  radius  of  wheel:  radius  of  axle. 

From  this  proportion  and 
the  one  above,  we  have  for  a 
perfect  wheel  and  axle  — 

R:  E::  De:  Dr  or  EDe  =  RDr. 

That  is,  the  wheel  and  axle, 
like  the  lever,  transmits  ener- 
gy without  loss  or  gain,  except 
in  so  far  as  there  is  loss  due  to 
friction.  The  loss  usually  amounts  to  from  10%  to  20%, 
depending  principally  upon  the  condition  of  the  bearings. 

Applications.  —  The  applications  of  the  wheel  and  axle, 
in  various  modified  forms,  are  numerous  and  important. 
The  capstan  (Fig.  140),  used  on  small  vessels  for  raising 
and  lowering  the  anchor,  has  a  vertical  axle  or  drum,  and 
the  effort  is  applied  at  the  end  of  hand  spikes,  inserted  in 
holes  at  the  top.  Ratchets  at  the  bottom  prevent  backward 
motion  during  any  interruption  of  the  work.  The  capstan 
or  windlass  is  used^in  connection  with  pulleys  for  moving 
houses,  and,  as  part  of  the  hoisting  tackle  of  derricks,  for 
lifting  or  lowering  heavy  weights  (Fig.  141).  Wheels 
of  unequal  size,  interacting  by  means  of  cogs  (Fig.  159)  or 
connected  with  a  belt  or  a  chain,  as  in  the  sewing-machine, 
lathe,  bicycle,  clock,  etc.,  are  further  applications  of  the 
same  principle,  the  change  of  speed  and  of  force  being 
always  in  inverse  proportion,  barring  losses  due  to  friction. 


170 


DYNAMICS 
PROBLEMS 


1.  (a)  The  radius  of  a  wheel  is  40  cm.  and  the  radius  of  the  axle  12  cm. 
Neglecting  friction,  what  effort  is  required  to  raise  a  load  of  150  kg.?  (b) 
Through  what  distance  does  the  effort  act  in  raising  the  load  35  m.?  (c)  How 


FIG.  141.  —  Compound  Windlass  on 
a  Derrick. 


FIG.  142. — Derrick,  with  Hoisting 
Tackle  of  Rope  and  Pulleys. 


much  work  is  done  by  the  effort?     (d)  How  much  is  done  against  the  weight 
of  the  load? 

2.  What  effort  will  be  required  to  raise  a  weight  of  200  kg.  with  a  wheel 
and  axle  the  efficiency  of  which  is  90%,  the  radius  of  the  wheel  being  42  cm. 
and  the  radius  of  the  axle  14  cm.? 

3.  The  weight  in  the  preceding  problem  is  raised  25  m.     Find   (a)  the 
work  done  upon  the  machine;    (b)  the  work  done  against  gravity;    (c)  the 
energy  wasted. 

4.  Find  the  mechanical  advantage  of  the  windlass  with  gear-wheels 
shown  in  Fig.  144,  if  the  length  of  the  crank  arm  is  16  in.,  the  radius  of  the 
axle  or  "barrel"  round  which  the  rope  is  wound  5  in.,  the  number  of  cogs 
on  the  small  wheel  15  and  on  the  large  wheel  75.     If  the  efficiency  of  the 
machine  is  75%,  what  effort  is  required  to  raise  1000  lb.? 

144.  The  Pulley.  —  A  pulley-block  or  tackle-block  holds 
from  one  to  six  pulleys,  which  are  capable  of  turning  indi- 
vidually* at  unequal  rates.  A  common  form  of  hoisting 
tackle  (Fig.  142)  consists  of  a  rope  and  two  pulley-blocks, 


MACHINES 


171 


FIG.  143. — The  Fixed 
Pulley. 


FIG.  144. — The  Mov- 
able Pulley. 


one  being  attached  to  a  fixed  support  and  the  other  to  the 
moving  object.  The  pulleys  of  a  fixed  block  are  called 
fixed  pulleys,  and  those 
of  a  movable  block; 
movable  pulleys. 

Mechanical   Advan- 
tage. —  A  fixed  pulley 
(Fig.   143)  may  be  re- 
garded  as    a    continu- 
ously acting  lever  of  the 
first  class,  whose  arms 
are  radii  of  the  pulley. 
The  arms  being  equal, 
the  static  effort  and  the 
resistance  are  also  equal. 
This  is  further  evident  from  the  fact  that,  except  for  a  slight 
possible  difference  due  to  friction,  the  parts  of  the 
rope  on  the  two  sides  of  the  pulley  must  be  under 
the  same  tension.     The  only  advantage 
gained  by  the  use  of  one  or  more  fixed  pul- 
leys is  a  change  in  the  direction  of  the 
effort  (Fig.  129). 

The  movable  pulley  (Fig.  144)  is,  in  prin- 
ciple, a  continuously  acting  lever  of  the 
second  class,  the  arm  of  the  effort  being  the 
diameter  of  the  pulley,  and  the  arm  of  the 
resistance  its  radius.  Hence  the  mechan- 
ical  advantage  of  a  single  movable  pulley 
is  two  (R:  E  =  2).  This  follows  also  from 
the  fact  that  the  two  equal  upward  pulls 
of  the  rope  on  the  two  sides  of  the  pulley  to- 
gether suppoj-t  the  load.  FlG.  I46. 


FIG.  145. 


172  DYNAMICS 

When  pulleys  are  used  in  combination  by  passing  a  single  rope 
alternately  round  the  pulleys  of  a  fixed  and  a  movable  block  (Figs. 
142,  145,  and  146),  the  mechanical  advantage  is  determined  by  assum- 
ing the  tension  to  be  the  same  in  all  parts  of  the  rope.  The  load  is 
thus  sustained  by  as  many  equal  parallel  forces  as  there  are  parts  of 
the  rope  running  to  and  from  the  movable  block;  and  each  of  these 
forces  is  equal  to  the  effort.  That  is,  if  n  denotes  the  number  of 
times  the  rope  passes  to  and  from  the  movable  block,  then  nE  =  R 
or  R:E  =  n,  and  the  mechanical  advantage  is  n. 

The  Law  of  Work.  —  When  a  load  is  moved  a  certain 
distance  Dr  with  a  rope  and  pulleys,  the  distance  between 
the  fixed  and  the  movable  pulley-blocks  decreases  by  an 
equal  amount,  and  hence  each  of  the  n  parts  of  the  rope 
extending  between  the  blocks  shortens  by  just  that  much. 
Since  the  whole  rope  is  of  fixed  length,  the  free  end  of  it, 
where  the  effort  is  applied,  moves  through  a  distance  De 
equal  to  n  times  the  resistance  distance,  or  nDr.  Stated 
the  other  way  about,  the  load  moves  through  a  distance 
only  one  n/A  as  great  as  that  through  which  the  agent  acts. 
Thus,  with  perfect  efficiency,  gain  of  force  by  means  of  a 
set  of  pulleys  is  secured  at  the  expense  of  a  proportionate 
loss  of  distance.  Expressed  mathematically, 

R:  E  =  n  and  De:  Dr  =  n; 
hence  R:  E::  De:  Dr,  or  EDe  =  RDr.  • 

Efficiency.  —  When  a  load  is  raised  by  means  of  a  rope 
thrown  over  a  beam,  the  effort  greatly  exceeds  the  resist- 
ance of  the  load,  on  account  of  the  sliding  friction  between 
the  rope  and  the  beam;  and  the  two  parts  of  the  rope  are 
thus  under  unequal  tension.  When  the  rope  is  passed  over 
a  pulley,  friction  is  greatly  reduced  but  not  wholly  elim- 
inated; hence  there  is  a  slight  difference  in  the  tension  of 
the  ropes  on  the  two  sides  of  the  pulley  when  work  is  in 
progress.  With  a  set  of  pulleys,  there  is  loss  of  tension 


MACHINES  173 

from  the  free  end  of  the  rope  to  the  fixed  end,  at  every  point 
where  it  passes  over  a  pulley;  hence  the  working  effort 
must  be  greater  than  the   average  tension  in   the   rope 
and  the  work  on  the  load  is 
necessarily  less  than  the  work 
of  the  agent.     The  usual  effi- 
ciency of   a  hoisting   tackle 
of  four  to  six  pulleys  is  from 
60%  to  75%. 

PROBLEMS 

1.  Draw  a  diagram  of  a  system 
of  pulleys  such  that  the  mechanical 

advantage  is  seven;   such  that  it  is     FIG.  147.  — A  Steering-wheel  and  Gear. 
•  ut  W,   the    wheel;    B,   the  barrel;    C, 

chains;    S,  S     standards;     R,     the 

2.  What  effort  is  required  to  raise         Rudder-head;  T,  the  tiller, 
a  load  of  500  Ib.  with  a  set  of  pulleys 

arranged  as  in  Fig.  146,  the  efficiency  being  70  %? 

3.  The  steering  wheel  and  gear  of  a  vessel  shown  in  Fig.  147  is  a  com- 
bination of  three  machines,  of  which  the  last,  T,  called  the  tiller,  is  rigidly 
attached  to  the  rudder-head,  R.     Describe  the  action  of  this  compound 
machine,  and  express  its  mechanical  advantage  in  terms  of  the  appropriate 
dimensions  of  its  parts. 

4.  Let  the  boys  of  the  class  investigate  and  report  on  the  mechanism 
of  a  bicycle,  with  special  reference  to  the  ratio  of  gain  of  speed;  and  let 
the  girls  prepare  a  like  report  on  the   mechanism    of   a  sewing-machine. 
Express  in  terms  of  the  determining  factors  the  distance  that  a  bicycle 
goes  with  each  revolution  of  the  pedals,  and  the  number  of  stitches  that  a 
sewing-machine  takes  for  each  up  and  down  motion  of  the  pedal. 

145.  The  Inclined  Plane.  —  The  principal  use  of  the 
inclined  plane  is  to  raise  heavy  bodies  that  can  be  rolled. 
When  a  barrel  of  flour  is  placed  in  a  wagon  by  rolling  it 
up  a  heavy  plank,  the  plank  is  used  as  an  inclined  plane. 

The  laws  of  the  inclined  plane  have  already  been  fully 
considered  (Arts.  74  and  133).  What  is  the  measure  of 
its  mechanical  advantage?  How  has  it  been  shown  that, 


174 


DYNAMICS 


FIG.  148. 


disregarding  friction,  the  work  of  the  agent  is  equal  to  the 
work  (against  gravity)  on  the  load? 

146.  The  Wedge.  —  The  wedge  is, 
in  principle,  a  movable  inclined  plane. 
It  is  used  in  separating  surfaces 
against  great  resistance,  as  in  split- 
ting logs  and  timbers  (Fig.  148).  The 
motion  of  a  wedge  is  opposed  by  great 
sliding  friction  against  its  faces,  and 
its  efficiency  is  consequently  low.  This 
friction  is  useful,  however,  as  it  keeps 
the  wedge  from  slipping  out  of  place  during  the  intervals 
between  the  blows  that  drive  it  farther  in.  The  wedge  in 
actual  use  departs  to  such  an  extent  from  the  conditions 
of  a  perfect  machine  that  a  study  of  the  ideal  relations  is 
scarcely  profitable.  It  is  obvious,  however,  that  the 
thinner  the  wedge  the  greater  is  its  mechanical 
advantage.  The  ax,  the  knife,  and  the  chisel  are 
forms  of  the  wedge  adapted  to  special  uses. 

147.   The    screw  is  a  cylinder  about  which  ex- 
tends a  spiral  projection  or  thread.     That  the  thread 
is  a  modified  form  of  the  inclined  plane  can  be  shown 
by  winding  a  paper  triangle  about  a  pencil  (Fig. 
149).     The  distance  between  adjacent  turns  of  the 
thread,  measured  parallel  to  the  axis,  is  called  the 
pitch  of  the  screw.     The  nut  in  which  the    screw 
turns  is  provided  with  a  spiral  groove  to  re- 
ceive the  thread. 


FIG.  149. 


The  lifting-jack  (Fig.  150)  is  a  screw  operated 
by  means  of  a  lever.  It  is  used  for  lifting  heavy 
bodies,  as  in  replacing  a  derailed  locomotive  or  in  raising  a  house. 
During  one  complete  turn  of  the  lever,  the  load  is  raised  a  distance 


MACHINES  175 

equal  to  the  pitch  of  the  screw,  and  the  effort  acts  through  the 
distance  2irA,  A  being  the  arm  of  the  lever.  Hence  the  work 
done  by  the  agent  during  one  turn  is  £  X  2irA,  and  the  work 
done  on  the  load  is  'Rp.  Assuming  an  efficiency  of  100%,  we  should 
have 

Q.TT  A 

E  X  2-irA  =  Rp,  or  R  :E  =  ——  -  (28) 

Thus  with  a  lever  arm  2  ft.  long  and  a  pitch  of  \  in.,  the  mechanical 
advantage  would  be  302.  But  the  friction  is  always  great,  in  fact 
more  than  sufficient  to  hold  the  screw  in  place  against  any  resistance. 
If  we  take  the  efficiency  as  i,  the  actual  mechanical  advantage  with 
the  above  dimensions  would  be  100;  under  which  conditions  an 
effort  of  100  Ib.  would  still  be  sufficient  to  raise  a  load  of  5  tons. 


FIG.  1 50. — Jack-screw.  FIG.  151.  —  Copying  Press. 

The  screw  propeller  (Fig.  162)  is  an  efficient  machine  for  doing 
work  on  a  large  scale.  In  many  applications  of  the  screw,  such  as 
the  copying  press  (Fig.  151)  and  the  vise,  its  purpose  is  to  exert  great 
static  pressures;  in  others,  as  wood-screws,  machine-screws,  and  bolts, 
its  use  is  to  hold  the  parts  of  bodies  together. 

PROBLEMS 

1.  A  block  at  rest  upon  a  board  100  cm.  long  begins  to  slide  when  the 
board  is  inclined  so  that  its  higher  end  is  40  cm.  above  the  other.  Friction 
is  what  percentage  of  the  pressure  of  the  block  on  the  plane?  (The  ratio 
of  the  sliding  friction  to  the  pressure  between  two  surfaces  is  called  the 
coefficient  of  friction  for  those  surfaces.) 


i76 


DYNAMICS 


2.   Assuming  an  efficiency  of  80%,  what  force  is  required  to  haul  a  load 
of  2  tons  (including  the  weight  of  the  wagon)  up  a  grade  such  that  the  ascent 
is  i  ft.  in  a  distance  of  10  ft.? 

3.  A  lifting-jack,  the  screw  of  which  has  a 
pitch  of  i  in.,  is  used  to  raise  a  load  of  5  tons. 
The  effort  is  applied  at  the  end  of  a  lever  3  ft. 
long.   The  efficiency  is  25%.     Find  the  effort. 

4.  The  worm-wheel  (Fig.  152)  consists  of 
an  endless  screw  or  worm  and  a  gear-wheel. 
The  worm  drives  the  wheel,  and  is  itself  driven 
by  a  crank  or  a  wheel.     Express  the  mechan- 
ical advantage  of  this  compound  machine  in 
terms  of  its  dimensions. 


FIG.  152.  —  Worm-wheel. 


5.  Show  that  the  hydraulic  press,  regarded  as  a  perfect  machine,  con- 
forms to  the  law  that  the  work  of  the  agent  is  equal  to  the  work  on  the  load. 

VI.  ENERGY 

148.  Energy.  —  The  kinetic  energy  of  a  body  may  be 
denned  as  the  measure  of  its  capacity  for  doing  work,  by 
virtue  of  its  mass  and  its  motion.  A  body  at  rest  may  or 
may  not  be  capable  of  doing  work.  A  charge  of  powder 
behind  a  rifle  ball  has  a  definite  capacity  for  doing  work 
upon  the  ball  which  a  mass  of  sand  in  the  same  situation 
would  utterly  lack.  A  bent  bow  can  do  work  upon  an 
arrow,  but  an  unbent  bow  can  not,  although  both  are  at 
rest.  A  locomotive  with  steam  up  has  a  certain  capacity 
for  doing  work,  which  we  know  resides  in  the  supply  of 
steam  in  the  boiler.  Examples  might  be  multiplied  indefi- 
nitely of  bodies  capable  of  doing  work,  but  not  in  motion. 
In  all  such  cases  the  body  is  said  to  possess  energy.  En- 
ergy, then,  depends  upon  various  conditions  and  properties 
of  bodies,  and  consequently  exists  in  various  forms. 

To  discuss  the  different  forms  of  energy  and  their  rela- 
tion to  one  another  would  be  nothing  less  than  to  summarize 
the  whole  of  physics.  The  fuller  knowledge  of  such  mat- 
ters must  therefore  come  in  the  regular  progress  of  the  sub- 


ENERGY  177 

ject.  Forms  of  energy  other  than  mechanical  are  referred 
to  in  the  present  chapter  only  because  they  obtrude  them- 
selves upon  our  attention  as  soon  as  we  begin  to  consider 
energy  at  all. 

149.  Mechanical  Energy.  —  In  bending  a  bow  work  is 
done  against  its  elastic  resistance,  and  the  result  is  a  state 
of  strain  (Art.  87)  or  a  distortion  of  the  elastic  body.  In 
recovering  from  distortion,  the  bow  can  do  work  equal  in 
amount  to  the  work  done  in  bending  it;  and  this  amount 
of  work  is  the  measure  of  its  energy  of  strain.  All  elastic 
bodies,  solid  or  fluid,  possess  energy  of  this  kind  when  in 
a  state  of  strain.  The  coiled  spring  of  a  watch  or  a  clock, 
the  compressed  air  in  an  air  rifle,  and  the  steam  in  an  engine 
boiler  are  familiar  examples. 

A  body  so  situated  that  it  is  capable  of  descending  to  a 
lower  level  possesses  a  definite  store  of  energy  by  virtue  of 
its  mass,  its  elevation,  and  the  earth's  attraction.  This  is 
commonly  known  as  energy  of  position.  It  is  measured 
by  the  product  of  the  weight  of  the  body  and  its  elevation. 

For  example,  if  the  hammer  of  a  pile  driver  weighs  1000  Ib.  and  is 
raised  to  a  height  of  25  ft.,  its  energy  of  position  is  25,000  ft.-lb.; 
for,  in  falling  from  that  height,  its  weight  does  25,000  ft.-lb.  of  work 
upon  it,  imparting  25,000  ft.-lb.  of  kinetic  energy  to  it,  and  enabling 
it  to  do  25,000  ft.-lb.  of  work  upon  a  pile  at  the  end  of  its  fall. 

Water  power  is  energy  of  this  sort,  and  is  utilized  by 
means  of  water-wheels  (Art.  157). 

Energy  of  strain  and  energy  of  position  are  two  vari- 
eties of  potential  energy.  In  more  definite  terms,  they  are 
the  two  mechanical  varieties  of  potential  energy,  for  poten- 
tial energy,  in  the  broadest  sense,  means  static  energy,  as 
distinguished  from  kinetic,  and  includes  such  forms  as  the 
energy  of  coal,  which  is  chemical  energy,  not  mechanical. 


178  DYNAMICS 

The  essential  conditions  for  potential  energy  of  a  mechan- 
ical nature  are  the  existence  of  force  tending  to  move  the 
body  or  to  cause  relative  motion  of  its  parts,  and  room  for 
such  motion  to  take  place. 

Energy  of  strain,  energy  of  position,  and  the  kinetic 
energy  of  moving  masses  are  the  different  forms  of  mechan- 
ical energy.  They  are  the  result  of  mechanical  work  (the 

only  kind  of  work  yet  con- 
sidered), and  they  are  avail- 
able for  doing  mechanical 
work. 

150.  Energy  of  Rotation.  Fly- 
wheels.—  The  kinetic  energy  of 
a  rotating  body  is  frequently  a 
matter  of  importance,  particu- 
larly in  the  operation  of  machin- 
ery. The  flywheel  of  an  engine 
(Fig.  153)  is  an  instructive  ex- 
ample. It  is  always  massive 
and  is  frequently  of  great  size; 
and,  when  in  rapid  motion,  it 

possesses  a  large  store  of  kinetic  energy.  This,  of  itself,  however,  is 
of  no  importance,  since  the  wheel  can  pay  out  no  more  energy 
than  it  receives  from  the  engine.  But  the  large  energy  capacity 
of  the  wheel  enables  it  to  pay  out  this  energy  at  a  practically  con- 
stant rate  throughout  each  revolution,  while  receiving  it  intermit- 
tently from  the  engine,  with  each  stroke  of  the  piston.  This  action 
changes  what  would  otherwise  be  an  unsteady,  jerky  mction  of  the 
engine  and  machinery  into  a  steady  motion;  just  as  a  water  tank, 
while  receiving  its  supply  in  spurts  from  an  ordinary  pump,  main- 
tains a  steady  flow  through  an  outlet  pipe. 

The  greater  the  kinetic  energy  of  a  flywheel  the  more  effective  will 
it  be  as  an  equalizer  of  motion.  Now  the  energy  of  the  wheel  as  a 
whole  is  merely  the  summed  up  energy  of  its  parts;  and  the  energy 
of  any  part,  of  mass  mi,  distributed  in  the  form  of  a  ring  of  radius  r\ 
(Fig.  153),  is  \  miViz  (Formula  21).  For  a  given  number  of  revolu- 


FIG.  153.  —  Flywheel. 


ENERGY  179 

tions  per  second,  Vi  is  proportional  to  the  radius  r\.  (Why?)  Hence 
the  energy  of  a  given  portion  of  the  mass,  as  mi,  is  proportional  to 
the  square  of  its  distance  from  the  axis.  It  follows  that  the  effective- 
ness of  a  flywheel  is  increased  by  having  as  much  of  its  mass  as  is 
possible  at  the  greatest  distance  from  the  axis,  i.e.  in  the  rim;  and 
the  larger  the  diameter  of  the  wheel  the  greater  will  be  the  advantage 
thus  gained. 

Since  kinetic  energy  is  proportional  to  the  square  of  the  velocity, 
the  energy  of  a  flywheel  increases  as  the  square  'of  the  number  of 
revolutions  per  second.  Hence  a  small  flywheel  is  effective  if  it  is 
run  at  high  speed. 

151.  Heat  and  its  Relation  to  Mechanical  Work.  —  Fric- 
tion generates  heat.  The  hands  are  warmed  by  rubbing 
them  briskly  together.  A  match  is  ignited  by  drawing  it 
rapidly  over  a  rough  surface.  The  coaster  brake  of  a 
bicycle,  when  applied  on  a  steep  grade,  quickly  becomes 
hot  enough  to  burn  the  fingers.  When  the  axle  of  a  car 
wheel  is  not  properly  lubricated  to  diminish  friction,  a 
"hot  box"  results,  and  the  temperature  may  even  rise 
sufficiently  to  set  the  car  on  fire.  When  a  cord  or  rope  is 
grasped  tightly  and  drawn  rapidly  through  the  hand,  the 
heat  generated  quickly  causes  a  burn.  When  a  tool  is 
ground  on  a  dry  grindstone,  it  becomes  too  hot  to  touch.  In 
most  cases  of  friction,  it  is  true,  there  is  no  noticeable  rise 
of  temperature;  but  this  is  due  to  the  fact  that  the  heat 
is  not  generated  rapidly  and  escapes  readily  to  surround- 
ing bodies.  The  energy  expended  in  overcoming  friction 
always  produces  heat.  In  fact,  the  energy  thus  expended 
becomes  heat;  for,  as  will  be  more  fully  explained  later, 
heat  is  a  form  of  energy,  and  the  amount  generated  by  fric- 
tion is  always  equivalent  to  the  amount  of  mechanical  work 
done  against  the  friction. 

When  a  moving  body  is  brought  to  rest  in  the  act  of 
imparting  motion  to  another  body,  its  kinetic  energy  is 


i8o  DYNAMICS 

transferred  to  the  other  body;  when  it  is  brought  to  rest 
by  friction,  its  kinetic  energy  is  transformed  into  heat, 
partly  within  itself  and  partly  within  the  body  or  bodies 
that  stop  it.  The  same  transformation  of  energy  occurs 
when  a  body  is  suddenly  stopped  by  impact  against  a  body 
which  it  does  not  move.  Thus  a  piece  of  lead  can  be 
noticeably  heated  by  rapidly  hammering  it  on  an  anvil, 
and  bullets  are  often  partly  melted  by  the  heat  generated 
when  they  strike  a  steel  target  or  a  stone. 

The  change  of  heat  into  mechanical  energy  which  is 
accomplished  by  means  of  steam  and  other  heat  engines 
is  further  evidence  that  heat  is  itself  a  form  of  energy. 
Whether  this  energy  is  kinetic  or  potential  and  in  what  it 
consists  are  questions  to  be  considered  in  a  later  chapter. 

152.  Other  Forms  of  Energy.  —  Heat  is  generated  dur- 
ing many  chemical  changes,  as  in  the  burning  of  any  fuel, 
the  decay  of  vegetable  matter,  the  slaking  of  lime,  etc. 
Burning  produces  light,  also  a  form  of  energy,  as  well  as 
heat.  The  chemical  changes  in  an  electric  battery  produce 
electrical  energy.  Substances  capable  of  generating  heat, 
light,  or  other  forms  of  energy  by  chemical  change  are  said 
to  possess  chemical  energy.  This  is  a  form  of  potential 
energy. 

All  the  movements  of  an  animal  involve  an  expenditure 
of  muscular  energy;  for  the  movements  are  due  to  muscu- 
lar action,  and  in  this  action  the  muscles  do  work.  The 
amount  of  potential  energy  stored  in  the  muscles  is  great; 
a  horse,  for  example,  can  do  nearly  two  million  foot-pounds 
of  work  per  hour  for  several  hours.  It  is  evident,  however, 
that  the  amount  is  not  unlimited,  for  any  animal  becomes 
exhausted  after  prolonged  exertion.  The  renewed  supply 
of  energy  comes  from  the  food  eaten.  Muscular  energy  is 


ENERGY  181 

available  for  doing  work  only  through  chemical  changes 
by  which  the  muscles  are  in  part  consumed,  much  as  fuel 
is  consumed  in  a  fire.  Similar  changes  take  place  in  all 
the  organs  of  the  body  while  they  -are  performing  their 
special  functions,  and  some  of  the  energy  is  always  liber- 
ated as  heat.  It  is  this  heat  that  maintains  the  temper- 
ature of  the  body. 

The  principal  forms  of  energy  with  which  physics  deals 
are  the  forms  of  mechanical  energy  already  considered, 
and  the  energy  of  heat,  sound,  light,  and  electricity. 

153.  The  Conservation  of  Energy.  —  Energy,  as  we  have 
learned,  is  capable  of  transference  from  one  body  to  an- 
other and  of  transformation  from  one  form  into  another. 
Either  of  these  changes  may  take  place  alone,  or  they  may 
occur  simultaneously.  Is  energy  capable  of  other  changes 
than  these?  Can  energy  cease  to  exist  as  energy,  either 
by  becoming  something  else  or  simply  by  ceasing  to  be? 
Is  it  possible  to  make  or  create  energy,  or  by  any  device 
to  increase  energy  as  we  do  force?  After  long-continued 
observation  and  experiment,  these  questions  were  all 
answered  in  the  negative  about  the  middle  of  the  last 
century,  and  later  advance  in  science  and  invention  has 
only  confirmed  the  answer.  All  experience  teaches  that 
some  portion  of  matter  has  lost  whatever  energy  another 
portion  of  matter  gains,  and  that  energy  in  any  form  dis- 
appears only  by  transformation  into  an  equivalent  amount 
of  energy  in  some  other  form  or  forms.  This  is  known  as 
the  law  or  principle  of  the  conservation  of  energy.  Briefly, 
it  asserts  that  energy  is  not  created  or  destroyed  in  any 
phenomenon  or  process  known  to  man.  Energy  is  in  this 
respect  like  matter;  the  total  quantity  of  either  in  the  uni- 
verse, so  far  as  we  know,  remains  constant. 


182  DYNAMICS 

Assuming  the  principle  of  the  conservation  of  energy,  we  can 
give  a  more  complete  account  of  phenomena.  The  following  cases 
will  serve  to  illustrate.  When  one  highly  elastic  body  strikes  another, 
the  greater  part  of  the  energy  continues  in  the  kinetic  form  in  one  or 
the  other  or  in  both  of  the  bodies,  depending  upon  conditions;  the 
remainder  is  transformed  into  heat  and  sound.  For  example,  when 
one  ivory  ball  strikes  another  of  the  same  size  squarely  (Fig.  3), 
there  is  a  very  nearly  complete  transfer  of  kinetic  energy  from  one 
to  the  other.  When  a  steel  or  an  ivory  ball  is  dropped  upon  the 
smooth,  flat  surface  of  a  heavy  block  of  steel  or  stone,  the  ball  is 
slightly  flattened  and  the  block  dented;  and  the  kinetic  energy  of 
the  ball  becomes  for  the  instant  potential  energy  of  strain  in  the  two 
distorted  bodies.  All  of  the  energy  has  been  transformed  by  the 
impact,  and  some  of  it  has  been  transferred  to  the  block.  In  the 
immediate  recovery  of  the  bodies  from  distortion,  the  energy  is 
again  changed  to  the  kinetic  form,  and  the  block  transfers  its  energy 
to  the  ball.  But  the  restoration  of  kinetic  energy  is  not  complete; 
for  in  both  the  impact  and  the  rebound  there  has  been  a  partial 
transformation  into  heat,  the  only  effect  of  which  is  an  imperceptible 
rise  of  temperature  in  both  bodies.  When  either  or  both  of  two  bodies 
is  inelastic  and  one  strikes  the  other,  the  kinetic  energy  is  largely  or 
wholly  converted  into  heat,  as  when  a  stone  falls  to  the  ground  or 
a  lead  bullet  strikes  a  stone.  The  melting  of  the  bullet  in  the  latter 
case,  as  sometimes  happens,  is  due  to  the  fact  that  the  amount  of 
energy  transformed  is  great  in  comparison  with  the  mass.  If  the 
earth  in  its  motion  round  the  sun  should  collide  "head  on"  with 
another  body  like  itself  and  moving  with  an  equal  velocity  in  the 
opposite  direction,  the  heat  generated  would  convert  the  entire  mass 
of  both  bodies  into  a  white-hot  vapor. 

154.  Availability  of  Energy.  —  While  energy  is  conserved 
in  all  its  changes,  it  tends  constantly  to  the  condition  of 
uniformly  diffused  heat,  in  which  form  it  is  no  longer 
available  to  man  for  doing  useful  work.  The  wasted  work 
of  machines  is  mechanical  energy  lost  as  heat  in  the  pro- 
cess of  transmission.  The  work  done  in  moving  wagons, 
street  cars,  and  trains  is  done  against  friction;  and  the 


DYNAMICS  OF  FLUIDS  183 

energy  thus  expended  is  dissipated  as  heat.  Coal,  oil, 
gas,  and  electricity  are  valuable  only  by  reason  of  their 
available  energy;  but,  once  used,  this  energy  passes  to  the 
unavailable  form.  Economy  of  energy  thus  becomes 
quite  as  important  a  matter  in  the  affairs  of  daily  life  as 
economy  of  materials. 

PROBLEMS 

1.  Show  that,  if  the  friction  of  the  air  is  negligible,  the  sum  of  the  kinetic 
and  potential  energies  of  a  body  thrown  vertically  upward  remains  constant 
during  its  flight. 

2.  Why  is  a  brake  not  heated  if  it  is  applied  with  such  force  that  the  wheel 
slides  along  the  ground  or  the  track?     Where  will  the  heat  then  be  generated? 

3.  What  are  the  principal  sources  of  energy  utilized  by  man,  and  how 
are  they  made  available? 

4.  Discuss  the  energy  of  a  swinging  pendulum. 

VII.   DYNAMICS  or  FLUIDS 

155.  Pressure  of  Moving  Fluids. — The  pressure  exerted 
by  a  moving  fluid  in  consequence  of  its  mass  and  its  motion 
must  be  distinguished  from  the  pressure  due  to  its  weight. 
The  gravity  pressure  of  the  air  is  between  14  and  15  Ib. 
per  square  inch,  or  over  2000  pounds  per  square  foot, 
whether  the  air  is  at  rest  or  in  motion.  This  is  a  balanced 
pressure.  It  does  not  bend  a  twig  or  disturb  the  most 
delicate  flower.  Air  in  motion  exerts  a  one-sided  pressure, 
which,  although  small  compared  with  its  gravity  pressure, 
is  nevertheless  responsible  for  all  the  familiar  effects  of 
winds. 

Experiments  have  shown  that  the  pressure  of  a  moving 
fluid  (water  or  air)  against  a  fixed  surface  varies  approxi- 
mately as  the  square  of  its  velocity.  The  pressure  of  the 
wind  against  a  fixed  surface  which  it  strikes  perpendicu- 
larly is  approximately  .004  V2  Ib.  per  square  foot,  V  being 


184  DYNAMICS 

the  velocity  of  the  wind  in  miles  per  hour.  In  computing 
the  necessary  strength  of  tall  buildings,  architects  allow 
for  a  maximum  wind  pressure  of  30  Ib.  per  square  foot, 
which,  by  the  above  formula,  corresponds  to  a  velocity 
of  nearly  90  mi.  per  hour.  Such  a  velocity  is  attained 
only  in  violent  and  destructive  storms. 

156.  Action  of  Wind  on  Sails.— When 
the  wind  strikes  a  surface  at  any  oblique 
angle,  the  force  that  it  exerts  is  perpendicu- 
lar to  the  surface,  except  for  the  tangential 
force  of  friction,  due  to  the  sliding  of  the  air 
along  the  surface.  This  tangential  force  is 
relatively  small,  and  may  be  disregarded. 
Let  WO  (Fig.  155)  represent  the  direction 
of  the  wind  against  the  sail  S  of  a  boat,  and 
OP  the  total  force  exerted.  OP  is  perpen- 
dicular to  the  sail,  and  is  equivalent  to  the  two  forces  OL  and 
OF.  The  sideward  component  OL  is  opposed  by  the  resistance 
of  the  water  against  the  broadside  of  the  boat,  and  produces 
little  effect.  The  forward  component  OF  is  effective  in  propelling 
the  boat.  It  is  thus  possible  for  a  vessel  to  sail  obliquely  against  the 
wind,  as  is  clearly  shown  in  the  diagram,  and,  by  tacking  (sailing 
first  to  one  side  of  the  wind,  then  the  other),  to  steer  a  general  course 
exactly  opposite  to  the  direction  of  the  wind. 

In  sailing  directly  with  the  wind  the  pressure  against  the  sail 
decreases  as  the  speed  of  the  vessel  increases;  and,  if  the  speed  should 
become  equal  to  that  of  the  wind,  the  pressure  would  vanish.  But 
some  pressure  is  always  necessary  to  maintain  the  motion  of  the 
vessel;  hence  so  great  a  speed  is  impossible.  On  the  other  hand,  in 
sailing  obliquely  toward  the  wind,  as  shown  in  the  figure,  the  wind 
would  overtake  the  sail  and  exert  pressure  upon  it  even  if  the  vessel 
was  traveling  much  faster  than  the  wind.  This  curious  result  is 
actually  attained  with  ice-boats,  which  require  little  force  to  keep 
them  going,  the  friction  between  their  steel  runners  and  the  ice  being 
slight.  A  record  of  90  mi.  an  hour  has  been  made  in  this  manner. 
The  action  of  the  wind  on  the  vanes  of  a  windmill  is  the  same  as 
upon  a  sail.  The  vanes  are  oblique  to  the  wind,  and  their  motion  is 


DYNAMICS  OF  FLUIDS 


185 


FIG.  156.— Overshot  Wheel. 


at  right  angles  to  it.     (Draw  a  diagram  similar  to  Fig.   155,  show- 
ing the  action  of  the  wind  on  a  vane,  and  explain  in  detail.) 

157.   Water-power.  —  The 

energy   of   running    water   is 

utilized  by  means  of  water- 
wheels  of  various  forms.     The 

overshot  wheel  (Fig.  156)  and 

the  undershot  wheel  (Fig.  157) 

are    the   earlier   and    simpler 

types.     They    are    still   used 

to  some  extent  where  only  a 

moderate  amount  of  power  is 

required    and    the  supply  of 

water    is    abundant    for    the 

purpose.     The  overshot  wheel 

requires    a   considerable   fall; 

and  in  order  to  utilize  all  the  available  power,  its  diameter 

must  be  equal  to  the  height  of  the  fall.     The  buckets  on 

the  circumference  of 
the  wheel  fill  with 
water  at  the  top  and 
empty  at  the  bot- 
tom. The  weight  of 
this  water  turns  the 
wheel.  The  under- 
shot wheel  is  used 
where  there  is  little 
fall.  The  boards  or 
buckets  which  project 
from  its  circumfer- 
ence at  regular  intervals  dip  into  the  running  water, 

and  are   driven  forward  by  the  current. 

The  available  power  of  a  stream,  expressed  in  foot-pounds 


FIG.  157.  — Undershot  Wheel. 


i86 


DYNAMICS 


per  second,  is  the  product  of  the  weight  of  water  supplied 
in  a  second  and  the  fall  or  "head"  which  can  be  utilized. 
(The  horse-power  is  equal  to  this 
product  divided  by  550.)  For 
a  given  power,  the  undershot 
wheel  takes  a  relatively  large 
flow  of  water,  as  its  efficiency  is 
only  about  25%.  Under  favor- 
able conditions  the  efficiency  of 
the  overshot  wheel  is  about  75%, 
but  the  height  of  fall  which  it 
can  utilize  is  limited  to  the  pos- 
sible diameter  of  the  wheel. 


FIG.  158.  —  Section  of  Turbine 
Wheel.    G,  guides;  B,  blades. 


158. 

reached 


Turbine  and  Pelton  Wheels.  —  Water-wheels  have 
a  high  degree  of  perfection  in  the  turbine  and  the 

Pelton  types.  In  the 
common  form  of  tur- 
bine wheel  (Figs.  158 
and  159)  the  water  is 
directed  inward  by  fixed 
guides,  G,  so  as  to  strike 
at  the  most  efficient  an- 
gle against  the  blades,  B, 
of  the  wheel.  The  wheel 
is  inclosed  in  an  outer 
case,  C  (Fig.  159),  to 
which  the  water  is  con- 
veyed from  a  higher 
level,  through  a  supply 
pipe  not  shown  in  the  fig- 
ure. This  pipe  is  joined 

FIG.  159.— Turbine  Water-wheel.  to  the   Case   On  the   front 


DYNAMICS  OF  FLUIDS 


side,  covering  the  large,  circular  opening.     The  water  fills 

the  supply  pipe  to  the  top;  hence  the  height  of  the  pipe 

determines  the  pressure 

at   the   wheel.  L  From 

the  outer  case  the  water 

forces  its  way  between 

the  guides,  G,  strikes 

against  the  blades  of 

the  wheel,  and  drops 

to  the  tail-race,  having 

expended  80%  or  more 

...  .  .  FIG.  1 60.  — Pel  ton  Wheel. 

of  its  energy  in  turning 

the  wheel.  The  shaft  of  the  turbine  is  sometimes  vertical, 
as  in  the  figure,  and  sometimes  horizontal.  The  turbine 
is  placed  above  ground  or  at  the  bottom  of  a  wheel  pit, 
depending  upon  the  location  of  the  power  house.  In  the 
latter  case  the  waste  water  finds  an  outlet  through  a  tun- 
nel, leading  from  the  bottom  of  the  pit. 

Turbine  wheels  utilize  the  available  power  to  the  full,  whether 
the  head  is  low  or  high,  for  the  greater  the  head  the  greater  will  be  the 
pressure  at  the  wheel.  The  turbines  of  the  Niagara  Falls  Power 
Company  are  located  at  the  bottom  of  a  wheel  pit  136  ft.  below  the 
level  of  the  supply.  The  shaft  of  each  turbine  extends  to  the  power 
house  above,  and  its  upper  end  carries  the  rotating  part  of  a  5000 
horse-power  dynamo.  The  turbines  of  the  Great  Western  Power 
Company  at  Big  Bend  on  the  Feather  River,  California,  work  under 
a  head  of  525  ft.,  and  each  develops  18,000  horse-power.  This  is 
the  greatest  head  which  has  been  utilized  by  means  of  the  turbine 
wheel,  up  to  the  present  time  (1911). 

The  Pelton  wheel  (Fig.  160)  is  especially  adapted  for 
utilizing  the  power  of  small  mountain  streams,  as  it  takes 
but  little  water  and  operates  under  any  head,  from  25  ft. 
up  to  the  highest  that  nature  provides.  The  water  is 


1 88  DYNAMICS 

conveyed  to  the  wheel  through  steel  pipes  capable  of  with- 
standing great  pressure,  and  issues  from  a  nozzle  in  a  small 
but  powerful  stream,  which  is  directed  against  the  lower 
buckets  of  the  wheel.  The  buckets  have  a  central  parti- 
tion, which  splits  the  stream,  deflecting  part  toward  each 
side.  The  water  is  thus  caught  and  held  by  the  buckets 
until  nearly  all  its  energy  is  imparted  to  the  wheel. 

Standard  Pel  ton  wheels  range  from  3  to  6  ft.  in  diameter;  but 
it  is  the  head  under  which  they  work,  rather  than  their  size,  that 
determines  their  power.  The  power  plant  of  the  Pike's  Peak  Hydro- 
electric Company  of  Colorado  Springs  utilizes  a  head  of  2150  ft., 
which  is  equivalent  to  a  pressure  of  935  Ib.  per  square  inch.  The 
water  issues  from  the  nozzles  with  a  velocity  of  22,300  ft.  per  min- 
ute, or  250  mi.  per  hour.  The  velocity  of  the  wheel  buckets  is  nearly 
half  as  great,  or  twice  the  velocity  of  the  fastest  express  train.  Each 
wheel  develops  1500  horse-power. 

The  development  of  electrical  science  within  the  past  thirty  years 
has  enormously  enhanced  the  industrial  importance  of  water-power. 
Formerly  water-power  could  be  utilized  only  at  its  source,  and  all 
but  an  insignificant  fraction  of  it  ran  to  waste  the  world  over.  It 
has  now  become  one  of  the  most  valuable  natural  resources.  In 
mountain  regions  where  high  heads  of  water  are  available,  electric 
power  stations  are  being  established  in  rapidly  increasing  numbers. 
Here  torrents  of  water  unceasingly  deliver  their  store  of  energy  to 
water-wheels,  and  these  to  dynamos  generating  electric  currents,  by 
which  the  energy  is  transmitted  to  distances  of  100  to  200  mi.  or 
more,  and  at  small  cost  in  most  cases  in  comparison  with  steam. 
"  The  immeasurably  vast  resources  of  power  available  by  this  means 
open  up  in  all  directions  new  fields  for  enterprise,  offering  profitable 
employment  for  both  labor  and  capital." 

159.  Resistance  of  the  Air  and  Water.  —  The  resistance 
of  still  air  to  the  motion  of-  a  body  through  it  varies  as  the 
square  of  the  velocity  of  the  body,  just  as  wind  pressure 
against  a  body  at  rest  varies  as  the  square  of  the  velocity 
of  the  wind  (Art.  155).  The  force,  whether  we  call  it 


DYNAMICS  OF  FLUIDS  189 

a  resistance  or  a  pressure,  is  determined  by  the  relative 
motion  of  air  and  body.  The  same  is  true  of  the  motion 
of  bodies  through  the  water. 

When  the  speed  of  a  train  is  not  over  5  mi.  per  hour, 
the  principal  resistance  to  its  motion  on  a  level  track  is 
friction  at  the  wheels;  when  its  speed  is  50  mi.  per  hour, 
the  principal  resistance  is  that  of  the  air,  which  is  100 
times  as  great  as  with  a  speed  of  5  mi.  per  hour,  while 
the  friction  at  the  wheel  is  practically  unchanged.  The 
high  speed  of  modern 
express  trains  empha- 
sizes the  importance  of 
diminishing  the  air  re- 
sistance as  much  as 
possible.  The  gasoline 
motor  car  shown  in  Fig. 


F'°'  rtl- 


Car  bui"  for 


161  represents  one  solu- 

tion  of  the  problem.    It 

carries  its  own  power,  and  its  wedge-shaped  end  divides 

the  air  as  the  prow  of  a  vessel  divides  the  water.     The 

resistance  of  the  air  is  turned  to  account  by  an  aeronaut 

when  he  drops  from  his  balloon  to  the  ground  with  the 

aid  of  a  parachute  (Fig.   112). 

Since  the  resistance  of  water  is  about  800  times  as  great 
as  that  of  air  (owing  to  its  greater  density),  the  proper 
design  of  vessels  is  a  matter  of  the  first  importance,  and  has 
long  been  the  subject  of  careful  investigation.  But  while 
a  good  design  diminishes  the  resistance  at  a  given  speed, 
it  does  not  alter  the  fact  that  the  resistance  increases  as 
the  square  of  the  speed. 

160.  Relation  of  Power  to  Speed.  —  The  work  done  in  propelling 
a  ship  is  the  product  of  the  resistance  to  the  ship's  motion  and  the 
distance  that  the  ship  travels.  The  work  done  in  a  second,  or  the 


i  go 


DYNAMICS 


power,  is  the  product  of  the  resistance  and  the  distance  traveled  in 
one  second.  But  the  resistance  varies  as  the  square  of  the  speed, 
and  the  distance  traveled  in  a  second  varies  as  the  speed;  hence 
the  power  necessary  to  propel  a  vessel  varies  as  the  cube  of  the 
speed.  To  double  the  speed  of  a  vessel  the  power  must  be 
increased  eight  fold;  to  increase  the  speed  from  20  knots  to  25  knots 
the  power  must  be  nearly  doubled  (25*  -5-  2o3  =  1.953). 

161.  The  Screw  Propeller.  —  The  power  of  engines  is  applied  in 
the  propulsion  of  ships,  tugs,  gasoline  launches,  etc.,  by  means  of 
the  screw  propeller  (Fig.  162).  The  mechanical  action  of  the  propel- 
ler is  exactly  opposite  to  that  of  the  windmill.  This  is  easily  seen  in 
the  case  of  the  electric  fan,  which  is  a  screw  propeller  designed  to 


FIG.  162.  —  Twin  Screw  Propellers      FIG.  163. — Motor  Ice-boat  propelled 
of  a  Motor  Boat.  by  an  Aerial  Screw. 

create  a  current  of  air.  A  windmill  would  accomplish  the  same  result 
if  it  were  run  by  an  engine.  But  while  the  fan  drives  a  current  of  air 
in  one  direction,  the  reaction  of  the  air  on  the  fan  tends  to  drive  it 
in  the  opposite  direction.  This  reaction  on  the  fan  is  transmitted  to 
the  bearings  of  the  axle,  and  thence  to  the  motor.  If  a  motor  and 
fan  are  mounted  on  a  light  carriage,  the  whole  becomes  a  self-pro- 
pelling machine,  running  in  one  direction  while  setting  up  a  current 
of  air  in  the  opposite  direction.  Similarly,  the  propellers  of  a  steam- 
ship drive  the  water  backward;  and  the  reaction,  transmitted  by  the 
shaft  and  its  bearings  to  the  vessel,  drives  the  vessel  forward. 

A  propeller  designed  to  work  in  water  usually  has  three  rounded 
blades,  as  shown  in  the  illustration.  An  aerial  screw,  such  as  is  used 
in  propelling  ice-boats  (Fig.  163),  dirigible  balloons  (Fig.  164),  and 
aeroplanes  (Figs.  165  and  166),  has  been  found  to  be  more  efficient 
when  constructed  with  only  two  blades  or  arms.  The  efficiency  of 


DYNAMICS  OF  FLUIDS 


igi 


a  ship's  propeller  is  generally  not  above  50%,  and  reaches  about  65% 
under  the  most  favorable  conditions.  Much  energy  is  necessarily 
lost  in  imparting  motion  to  the  water. 

162.    The   Navigation  of  the   Air.  —  The  well  known  spherical 
balloon,  invented  in  France  in  1773,  was  the  first  and,  until  recently, 


FIG.  164.  — The  Baldwin  Dirigible  Balloon. 

the  only  successful  device  for  navigating  the  air.  Balloons  of  this 
type  can  only  drift  with  the  wind.  The  utmost  that  the  aeronaut 
can  do  in  determining  his  course  is  to  choose  the  current  of  air  in 
which  his  balloon  shall  float.  This  he  does  by  throwing  out  ballast 
if  he  wishes  to  ascend,  or  by  opening  a  valve  to  let  some  of  the  gas 
escape,  if  he  wishes  to  reach  a  lower  level. 


UPPER  SUPPORTING  PLANE       SPROCKET  WHEELS  AND  CHA.NS 


ELEVATING  P 


SUPPORTING  PLANE 


EXIBLE'END 


FIG.  165.  —  The  Original  Wright  Biplane.     (The  first  successful  flying  machine.) 

The  dirigible  (steerable)  balloon  or  airship  is  an  invention  of 
recent  years,  and  is  still  being  improved.  Buoyancy  is  secured  by 
means  of  a  gas  bag,  as  in  the  older  type  of  balloon;  but  the  bag  is 
long  and  pointed,  to  diminish  air  resistance  (Fig.  164).  Motive 
power  is  provided  by  a  gasoline  engine,  operating  one  or  more  screw 
propellers.  The  steering  gear  includes  a  horizontal  rudder,  placed 


1 92  DYNAMICS 

in  front,  for  steering  up  or  down,  and  a  vertical  rudder  at  the  rear, 
for  steering  to  the  right  or  left.  A  car  or  a  long,  rigid  framework 
attached  to  the  under  side  of  the  gas  bag,  carries  the  engine  and 

propellers,  together 
with  tne  aeronaut, 
passengers,  and 
other  load.  The 
rigid  framework  of 
the  Baldwin  dirigi- 

FIG.  i66.-The  Bleriot  Monoplane.  (The  aeroplane  ble  shown  in  the 
in  which  M.  Louis  Bleriot  crossed  the  English  Chan-  figure  also  carries 
nel  from  Calais  to  Dover,  July  25,  1909.)  the  rudders.  This 

military  dirigible  was  built  by  Capt.  Thomas  A.  Baldwin  for  the 
United  States  Government  in  1908.  It  is  94  ft.  long  and  20  ft.  in 
diameter,  is  inflated  with  hydrogen,  and  carries  two  men.  Several 
dirigible  balloons  about  450  ft.  in  length  and  45  ft.  in  diam- 
eter, and  having  a  carrying  capacity  of  12  to  20  persons,  have 
been  built  and  successfully  operated  by  Count  Zeppelin,  of  Germany. 
These  huge  airships  have  a  car  at  the  front  and  another  at  the 
rear,  with  an  engine  in  each,  and  each  engine  drives  two  pro- 
pellers. "  Zeppelin  IV,"  which  was  destroyed  by  accident  during 
its  first  long-distance  flight,  maintained  an  average  speed  of  34  mi. 
per  hour. 

The  latest  form  of  aerial  craft  is  the  heavier-than-air  flying  ma- 
chine. Having  no  gas  bag,  these  machines  are  supported  in  their 
flight  only  by  the  reaction  of  the  air,  like  a  bird  or  a  kite.  The  only 
type  of  flying  machine  which  has  proved  successful  up  to  the  present 
time  (1911)  is  the  aeroplane.  Aeroplanes  are  classed  as  biplanes  and 
monoplanes,  the  former  having  two  principal  planes  or  supporting 
surfaces  (Fig.  165),  the  latter  only  one  (Fig.  166).  The  biplane,  as 
originally  designed  by  the  brothers  Wilbur  and  Orville  Wright, 
has  three  pairs  of  planes,  constructed  of  a  light  wooden  framework 
covered  with  muslin.  The  large  central  pair  are  the  principal 
supporting  planes.  They  are  slightly  convex,  viewed  from  above, 
and  slope  downward  toward  the  rear.  As  the  machine  is  driven 
forward  in  its  flight,  the  air,  striking  the  under  side  of  the  planes, 
exerts  an  upward  pressure  upon  them.  This  pressure  does  not 
exceed  two  or  three  pounds  per  square  foot  of  surface  when  the 
machine  is  carrying  two  men.  The  horizontal  planes  at  the  forward 


DYNAMICS  OF  FLUIDS  193 

end  serve  as  a  rudder  to  direct  the  machine  upward  or  downward 
in  its  flight.  (The  forward  planes  were  discarded  in  1910,  in  favor 
of  a  single  horizontal  steering  plane  at  the  rear.)  The  pair  of 
vertical  planes  at  the  rear  serve  as  a  rudder  for  horizontal  steering, 
like  the  rudder  of  a  ship.  The  machine  is  driven  by  a  light  but 
powerful  gasoline  engine,  working  two  aerial  screws. 

The  success  already  achieved  has  led  to  the  most  extravagant 
expectations  concerning  the  future  of  aerial  navigation.  Dirigible 
balloons  and  aeroplanes  are  now  regarded  as  a  necessary  part  of 
the  military  equipment  of  the  great  nations  of  the  world,  for  use  in 
scouting  and  carrying  despatches.  It  is  highly  probable  that  they 
may  render  important  service  in  other  limited  and  special  fields, 
as  in  the  exploration  of  regions  where  travel  on  land  is  beset  with 
great  difficulties  or  dangers.  But  no  form  of  aerial  craft  can  ever 
serve  as  a  practical  means  of  transportation  and  travel  in  the 
ordinary  circumstances  of  life.  The  Zeppelin  airship  is  nearly  as 
large  as  an  ocean  liner,  yet  its  carrying  capacity  is  no  more  than  half 
that  of  an  ordinary  street  car.  It  is,  moreover,  only  a  fair-weather 
vehicle,  so  far  as  safety  in  launching  and  landing  is  concerned. 
When  at  anchor  in  even  a  moderately  strong  wind  it  is  helplessly 
buffeted  about,  owing  to  the  enormous  area  subjected  to  wind 
pressure.  The  ordinary  carrying  capacity  of  the  largest  heavier- 
than-air  flying  machines  is  at  present  three  men;  and  this  is 
doubtless  very  near  the  possible  limit.  (Twelve  have  been  carried 
for  a  short  distance.)  But  the  one  decisive  factor  in  limiting  their 
field  of  usefulness  is  that  they  are  and  must  always  remain  peculiarly 
hazardous. 

PROBLEMS 

1.  Assuming  that  the  power  necessary  to  propel  a  steamship  varies  as 
the  cube  of  the  speed,  what  is  the  relation  between  the  speed  and  the  energy 
expended  in  a  given  time?  between  the  speed  and  the  energy  expended  in  a 
given  distance? 

2.  A  boy  rides  a  wheel  at  the  rate  of  1 2  mi.  per  hour  in  a  wind  of  8  mi.  per 
hour.      What  is  the  ratio  of  the  air  resistance  against  him  when  he  is  riding 
against  the  wind  to  the  resistance  when  he  is  riding  with  the  wind? 

3.  Discuss  the  action  of  a  ship's  rudder.     Discuss  the  action  of  the  for- 
ward horizontal  rudder  of  a  dirigible  balloon  or  an  aeroplane. 


CHAPTER  VII 
THE  MOLECULAR  THEORY  OF  MATTER 

163.  Introduction.  —  In  the  study  of  mechanics  we  have 
become  acquainted  with  some  of  the  most  important  phys- 
ical properties  of  matter  and  with  the  laws  resulting  from 
them.  We  have  learned  that  all  matter  occupies  space 
to  the  exclusion  of  other  matter,  that  it  possesses  inertia 
or  mass,  is  capable  of  storing  energy  in  various  forms, 
exerts  an  attractive  force  called  gravitation,  and  possesses 
elasticity  of  volume.  These  general  properties  of  matter 
are  the  material  basis  of  the  general  laws  and  principles  of 
mechanics,  such  as  Newton's  laws  of  motion  and  the  law 
of  gravitation.  We  have  seen  that  the  special  or  charac- 
teristic properties  that  distinguish  the  three  states  of 
matter  from  one  another  give  rise  to  other  less  general 
laws  and  principles,  as  presented  in  the  mechanics  of 
solids,  the  mechanics  of  liquids,  and  the  mechanics  of 
gases.  In  all  this  work  we  have  taken  the  properties 
of  matter  for  granted,  as  facts  of  observation  and  experi- 
ment; we  have  not  attempted  to  account  for  these 
properties. 

But  scientific  inquiry  does  not  end  with  the  attainment 
of  such  results  as  these,  important  as  they  are.  Laws  and 
principles  relate,  as  it  were,  only  to  the  surface  of  things. 
The  question  still  remains:  Why  are  the  facts  thus  and  so? 
Gases  exert  pressure  and  tend  to  expand  indefinitely;  but 
why  do  they?  What  is  the  minute  invisible  structure  or 
the  internal  condition  of  a  gas  which  causes  this  behavior? 

194 


THE  MOLECULAR  THEORY  OF  MATTER    195 

In  short,  what  is  a  gas?  Liquids  are  slightly  compress- 
ible, despite  the  fact  that,  even  under  the  microscope, 
they  seem  to  be  absolutely  continuous  bodies,  completely 
filling  the  space  they  occupy.  We  can  easily  see  how  a 
mass  of  loose  earth  or  a  piece  of  bread  can  be  pressed  into 
smaller  compass;  but  how  does  it  happen  that  liquids  are 
compressible  at  all?  Why  does  a  solid,  a  liquid,  or  a  gas 
expand  when  heated  and  contract  again  when  cooled? 
How  does  heat  convert  ice  into  water  and  water  into  invis- 
ible vapor?  To  answer  such  questions  as  these  about  the 
effects  of  heat,  we  must  not  only  know  more  about  matter, 
we  must  know  what  heat  itself  is.  Heat  is  a  form  of  energy, 
we  have  learned,  but  what  form?  In  what  does  it  con- 
sist? 

These  are  only  a  few  of  the  questions  that  arise  con- 
cerning the  physical  properties  of  matter  and  physical 
phenomena;  and  the  science  of  chemistry  presents  an 
equally  formidable  list,  relating  to  chemical  properties  and 
chemical  phenomena.  To  answer  such  questions  we  must 
know  what  matter  is  in  its  minutest  structure ;  and  the  eye 
is  hopelessly  incapable  of  giving  us  this  information,  even 
with  the  aid  of  the  most  powerful  microscope.  Similar 
difficulties  confront  the  investigator  in  all  departments  of 
science,  and  they  are  always  met  in  the  same  way.  Where 
direct  and  final  information  fails,  the  investigator  tries  to 
imagine  a  cause  that  would  give  rise  to  the  known  results. 
The  outcome  of  such  a  procedure  is  a  theory  or  possible 
explanation  of  the  facts  under  consideration.  In  trying 
to  account  for  the  facts  presented  in  this  chapter,  physi- 
cists and  chemists  have  formulated  the  molecular  theory  of 
matter  and  the  kinetic  theory  of  heat.  That  part  of  the  the- 
ory of  matter  which  relates  to  gases  is  known  as  the  kinetic 
theory  of  gas. 


196         THE   MOLECULAR  THEORY  OF  MATTER 

A  theory  is  not  necessarily  true  merely  because  it  affords  a  satis- 
factory explanation  of  all  the  known  facts  to  which  it  relates;  for 
it  is  conceivable  that  the  true  cause  may  be  very  different  from  the 
one  suggested.  Indeed,  it  has  happened  repeatedly  in  the  history 
of  science  that  rival  theories  have  been  ably  defended  at  the  same 
time  by  different  scientists  of  recognized  authority.  If,  at  any  time, 
a  new  fact  is  discovered  which  is  inconsistent  with  a  theory,  the 
theory  must  be  modified  to  bring  it  into  agreement  with  the  fact, 
or,  if  this  is  impossible,  it  must  be  abandoned.  Newly  discovered 
facts  have  often  served  to  distinguish  between  a  true  theory  and  a 
false  one.  The  ancients  believed  that  the  earth  was  fixed  in  space, 
and  that  the  apparent  motions  of  the  heavenly  bodies  were  real. 
Their  theory  of  the  universe  was  in  satisfactory  agreement  with  the 
facts  of  astronomy  then  known;  but  in  the  light  of  present  knowledge 
such  a  theory  would  be  absurd.  Only  a  few  of  the  scientists  of  a 
hundred  years  ago  dissented  from  the  opinion  then  generally  held 
that  heat  is  a  form  of  matter  without  weight.  This  opinion  was 
reasonable  then;  it  could  not  now  be  entertained  for  a  moment  by 
any  intelligent  person  acquainted  with  the  facts  that  have  since  been 
established. 

All  the  facts  that  a  theory  serves  to  explain,  taken  together, 
make  up  the  evidence  in  favor  of  its  truth.  If  this  evidence 
finally  becomes  conclusive,  as  knowledge  of  the  subject  increases, 
the  theory  becomes  an  established  fact.  The  Copernican  theory 
of  the  solar  system  was  true  in  the  main;  but  the  observations  of 
Tycho  Brahe  and  the  mathematical  researches  of  Kepler  and  Newton 
were  necessary  to  correct  and  complete  it.  It  then  became  an 
accepted  body  of  scientific  knowledge. 

I.   THE  STRUCTURE   OF  MATTER 

164.  The  Physical  Unit  of  Matter.  —  Any  substance  can 
be  cut,  broken,  or  otherwise  separated  into  parts,  each  of 
which  can  be  separated  into  smaller  parts,  and  so  on. 
Can  the  subdivision  of  any  kind  of  matter  be  continued 
indefinitely,  or  does  it  finally  come  to  a  definite  end?  A 
brittle  solid  can  be  crushed  or  ground  to  powder  of  such 
fineness  that  the  individual  particles  are  barely  visible 


THE  STRUCTURE   OF   MATTER  197 

under  the  most  powerful  microscope;  and  such  particles  are 
less  than  one  millionth  as  large  as  the  smallest  that  can 
be  seen  with  the  unaided  eye.  But  subdivision  into  even 
smaller  particles  than  these  is  of  common  occurrence.  A 
coin  is  visibly  worn  away  after  years  of  service;  a  knife 
loses  its  edge  from  continued  use ;  and  a  razor  is  noticeably 
dulled  in  shaving  once  with  it.  No  microscope  is  capable 
of  revealing  the  particles  lost  at  any  time  in  such  cases. 
Still  we  have  not  reached  a  limit  of  divisibility.  Water, 
standing  in  an  open  vessel,  slowly  disappears  by  evapora- 
tion ;  but  it  continues  to  exist  as  water  in  the  form  of  vapor, 
widely  diffused  in  the  air.  A  minute  fragment  of  musk 
will  continue  for  years  to  fill  a  room  with  its  odoriferous 
particles,  and  at  the  end  of  that  time  will  scarcely  be 
diminished  in  weight. 

It  would  be  impossible  to  determine  from  such  facts 
as  these  whether  there  is  a  limit  to  the  divisibility  of  a 
substance;  but  the  chemist  informs  us  that  there  is  such  a 
limit  —  that  there  is,  in  fact,  such  a  thing  as  the  smallest 
possible  particle  of  any  substance.  The  facts  that  lend 
the  strongest  support  to  this  conclusion  belong  to  the  sci- 
ence of  chemistry,  and  can  not  be  considered  here.  The 
smallest  possible  particle  of  a  substance  (other  than  an 
element)  is  called  a  molecule.  All  molecules  of  the  same 
substance  are  exactly  equal  in  size  and  weight,  and  are 
alike  in  every  respect;  but  the  molecules  cf  different  sub- 
stances are  unlike.  The  molecule  is  the  physical  unit  of 
matter;  it  is  the  limit  of  physical  divisibility.  In  all  purely 
physical  changes  the  molecule  preserves  its  identity. 
When  a  grain  of  sugar  dissolves  in  water,  the  molecules 
of  which  it  is  composed  are  separated  from  one  another, 
but  each  of  them  exists  as  a  molecule  of  sugar  in  the  solu- 
tion. A  molecule  of  ice  exists  as  the  same  identical  mole- 


198    THE  MOLECULAR  THEORY  OF  MATTER 

cule  after  the  ice  is  melted  and  the  water  evaporated.  The 
size  of  molecules  can  only  be  roughly  estimated;  but, 
judging  from  all  the  evidence,  there  must  be  millions  of 
them  in  the  smallest  microscopic  particle. 

165.  The  Chemical  Unit  of  Matter.  —  Since  the  molecules  of  one 
substance  differ  from  those  of  another,  the  molecules  themselves 
must  change  whenever  a  change  of  substance  takes  place,  as  is 
the  case  in  every  chemical  process.  For  example,  when  alcohol 
burns  it  unites  with  oxygen  from  the  air,  forming  carbon  dioxide 
(an  invisible  gas)  and  water  vapor.  (The  water  condenses  as  visible 
moisture  on  the  inner  surface  of  a  glass  jar,  held  inverted  over  the 
flame.)  In  this  process  two  substances  unite  chemically  to  produce 
two  other  substances.  The  molecules  of  alcohol  and  oxygen  are 
broken  up,  and  their  constituents  unite  to  form  molecules  of  carbon 
dioxide  and  water. 

Chemical  change  thus  involves  particles  of  matter  which  are  smaller 
than  molecules,  namely,  the  constituent  parts  of  the  molecules  them- 
selves; and  the  facts  of  chemistry  are  explainable  only  upon  the  sup- 
position that  these  smaller  particles  or  atoms,  as  they  are  called, 
continue  in  existence  unchanged  in  all  chemical  as  well  as  in  all 
physical  processes.  The  atom  is  thus  the  chemical  unit  of  matter. 

The  molecule  of  common  (ethyl)  alcohol  consists  of  two  atoms  of 
carbon,  six  atoms  of  hydrogen,  and  one  atom  of  oxygen,  and  its 
chemical  formula  is  CzHbOH.  In  such  formulas  the  initial  letter 
indicates  the  kind  of  atom,  and  the  figure  placed  after  and  a  little 
below  it  the  number  of  such  atoms  present  in  a  molecule.  A  letter 
without  a  figure  represents  one  atom.  The  oxygen  molecule  con- 
sists of  two  atoms  of  oxygen  (Oz)',  the  carbon-dioxide  molecule,  of 
one  atom  of  carbon  and  two  atoms  of  oxygen  (COz) ;  and  the  water 
molecule,  of  two  atoms  of  hydrogen  and  one  atom  of  oxygen  (HzO). 
In  burning,  one  molecule  of  alcohol  and  three  molecules  of  oxygen 
unite  to  form  two  molecules  of  carbon  dioxide  and  three  molecules 
of  water.  This  is  expressed  by  the  chemical  equation 

CzH.OH  +  zOz  =  2C02  +  3  H20, 

in  which  the  numerical  coefficients  denote  the  number  of  molecules. 
The  two  sides  of  a  chemical  equation  must  show  an  equal  number  of 


THE  STRUCTURE   OF  MATTER  199 

each  kind  of  atom,  since  no  atoms  are  created,  destroyed,  or  changed 
into  anything  else. 

166.  Elements  and  Compounds.  —  Any  substance  whose  mole- 
cules contain  unlike  atoms  is  called  a  chemical  compound.     Alcohol, 
carbon  dioxide,  and  water  are  examples.     Compounds  are  numbered 
by  the  thousands.     In  fact,  nearly  all  substances  belong  to  this  class. 

By  various  methods  all  chemical  compounds  can  be  separated  into 
their  constituents.  Substances  are  thus  obtained  which  resist  all 
attempts  to  decompose  them  further,  and  these  are  called  elements. 
The  presumption  is  that  the  molecules  of  an  element  really  consist 
of  but  one  kind  of  atom.  Practically  the  only  doubtful  cases  are 
some  of  the  newer  and  rarer  elements. 

The  elements  found  in  greatest  abundance  in  rocks  and  soils  are 
oxygen,  silicon,  aluminum,  iron,  calcium,  magnesium,  sodium,  and 
potassium.  The  principal  elements  composing  plant  and  animal 
tissues  are  carbon,  hydrogen,  oxygen,  and  nitrogen,  together  with 
small  quantities  of  sulphur,  phosphorus,  potassium,  sodium,  and 
magnesium.  Air  is  a  mixture  (not  a  compound)  of  oxygen,  nitrogen, 
a  little  argon,  still  less  of  carbon  dioxide,  and  a  varying  proportion 
of  water  vapor.  Water  is  composed  of  hydrogen  and  oxygen,  and 
table  salt  of  sodium  and  chlorine.  The  metals  iron,  aluminum, 
copper,  zinc,  tin,  lead,  mercury,  silver,  gold,  and  platinum  are  ele- 
ments; brass,  bronze,  and  German  silver  are  alloys  or  mixtures  of 
various  metallic  elements.  In  all  about  eighty  elements  are  known. 
"  It  is  impossible  to  state  the  number  precisely,  because,  owing  to  the 
great  rarity  of  some  of  them  and  the  imperfections  of  our  methods, 
there  are  always  some  whose  elementary  character  is  in  doubt." 
Eighteen  elements  alone  make  up  about  99%  of  the  earth's  crust, 
the  ocean,  and  the  atmosphere,  taken  together. 

167.  Molecular  Attraction.  —  The  strength  of  a  solid 
is  due  to  the  attraction  of  its  molecules  for  one  another. 
This  attraction  is  called  cohesion.     The  pieces  of  a  brittle 
solid,  as  glass  or  china,  do  not  unite  again  when  they  are 
fitted  accurately  to  each  other  and  pressed  firmly  together. 
This  shows  that  cohesion  acts  .only  at  exceedingly  minute 
distances.     In  fact,  its  greatest  range  is  estimated  at  .00005 
mm.  (two  million ths  of  an  inch). 


200         THE   MOLECULAR  THEORY  OF  MATTER 

t 

That  cohesion  only  requires  close  contact  is  readily 
shown  by  means  of  two  pieces  of  clean  plate  glass.  .When 
pressed  firmly  together,  they  cohere  with  sufficient  force 
to  sustain  the  weight  of  one  of  them.  Cohesion  plates  of 
metal,  having  surfaces  accurately  planed  and  polished,  give 
the  same  results.  Fragments  of  a  plastic  solid,  as  soft 
clay  or  putty,  unite  perfectly,  when  pressed  together, 
because  their  surfaces  are  brought  into  intimate  contact. 
In  the  process  of  welding,  the  two  pieces  of  metal  to  be 
united  are  rendered  plastic  by  heating.  Their  surfaces 
are  then  brought  within  the  range  of  cohesion  by 
hammering. 

Cohesion  in  liquids  is  shown  by  the  fact  that  their  par- 
ticles cling  together  in  drops  and  in  the  form  of  thin  films, 
as  in  soap  bubbles.  But  the  ease  with  which  a  liquid  is 
broken  up  into  drops  is  determined  by  the  freedom  of 
movement  of  the  molecules  (mobility)  rather  than  by  the 
strength  or  weakness  of  cohesion.  Experiments  have  shown 
that  water,  from  which  the  dissolved  air  has  been  driven 
by  boiling,  is  capable  of  sustaining  a  tensile  stress  of  70  Ib. 
per  square  inch,  while  clinging  to  the  walls  of  a  clean  glass 
tube.  In  general,  however,  cohesion  in  liquids  is  much 
weaker  than  in  solids. 

The  tendency  of  gases  to  expand  was  formerly  regarded 
as  evidence  that  the  molecules  of  a  gas  repel  one  another. 
Omitting  reasons  for  the  present,  we  may  say  that  there  is 
no  molecular  repulsion,  neither  is  there  cohesion  in  a  gas 
until  it  closely  approaches  condensation  into  a  liquid. 

168.  Adhesion.  —  There  is  no  evidence  of  any  difference 
in  the  nature  of  the  attraction  between  molecules  of  the 
same  kind  and  the  attraction  between  molecules  of  differ- 
ent kinds;  but  the  former  is  generally  called  cohesion  and 


THE  STRUCTURE  OF  MATTER  201 

the  latter  adhesion.     The  distinction  is  convenient  rather 
than  important. 

Adhesion  between  solids  is  very  common.  The  adhesive 
power  of  mud  is  sufficiently  familiar.  Butter  adheres  to 
a  knife  and  to  the  bread  upon  which  it  is  spread.  The 
adhesion  of  metals  is  utilized  in  gold  and  silver  plating. 
Ordinarily  there  is  no  adhesion  between  solids  when  brought 
in  contact;  but  this  is  only  because  their  surfaces  are  not 
sufficiently  close  together. 

Adhesion  between  solids  and  liquids  is  also  common. 
In  most  cases,  when  a  liquid  and  a  solid  are  brought  in 
contact,  the  liquid  clings  to  the  solid  and  wets  it.  This  is 
because  adhesion  between  the  two  is  greater  than  cohe- 
sion within  the  liquid.  The  surface  of  the  solid,  by  its 
superior  attraction,  tears  a  thin  layer  of  the  liquid  from 
the  remainder.  But  water  does  not  wet  a  surface  covered 
with  grease  or  wax,  and  mercury  wets  but  few  substances. 
In  such  cases  the  liquid  holds  to- 
gether in  somewhat  flattened  drops 
upon  the  surface  of  the  solid  (Fig. 

r  FIG.  167. 

167).    This  behavior  does  not  prove 

that  adhesion  is  wanting,  but  that   cohesion  within  the 

liquid  is  the  greater  of  the  two  attractions. 

Gases  also  adhere  to  solids,  penetrating  their  pores  and  forming 
a  very  thin  layer  upon  their  surface.  In  setting  up  a  barometer 
air  adheres  to  the  inner  wall  of  the  tube,  and  is  driven  off  only  by 
heating  the  mercury  till  it  boils. 

169.  Cohesion  and  Gravitation  Compared.  —  We  know  the  law 
of  gravitation,  but  not  its  cause.  We  know  neither  the  law  nor  the 
cause  of  cohesion;  but  it  is  evident  that  the  law  is  very  different 
from  that  of  gravitation,  for  cohesion  acts  only  at  insensible  distances, 
and  within  such  distances  it  is  enormously  stronger  than  gravitation 
between  the  same  masses.  Hence  the  strength  of  bodies  in  general 
depends  practically  entirely  on  cohesion;  and  gravitation  becomes 


2O2 


THE  MOLECULAR  THEORY  OF  MATTER 


appreciable  only  in  bodies  of  very  great  size.  The  strength  of  the 
earth  as  a  globe  depends  almost  wholly  on  gravitation.  If  we  im- 
agine the  earth  to  be  divided  into  hemispheres  by  any  plane  through 
its  center,  the  gravitational  attraction  by  which  the  hemispheres  are 
held  together  is  one  hundred  times  as  great  as  cohesion  would  be  if  the 
earth  were  made  of  solid  steel.  If  there  were  a  planet  fifty  miles 
in  diameter  having  the  same  density  as  the  earth  and  the  cohesive 
strength  of  sandstone,  gravitation  and  cohesion  would  be  equally 
effective  in  keeping  it  together. 

II.  MOLECULAR  PROPERTIES  OF  GASES 

170.  Diffusion  of  Gases.  —  If  two  bottles  containing 
different  gases  are  placed  one  over  the  other  and  mouth  to 
mouth  (Fig.  168),  with  the  denser  gas  at  the 
bottom,  and  are  left  standing  thus  for  half  an 
hour,  they  will  be  found  to  contain  an  equal 
mixture  of  the  two  gases.  This  can  be  readily 
shown  if  the  lower  bottle  is  filled  with  oxygen 
and  the  upper  with  coal  gas;  for  a  mixture  of 
these  gases  is  explosive,  and  an  explosion  occurs 
when  a  lighted  match  is  held  at  the  mouth  of 
either  bottle.  The  mixing  would  take  place 
much  more  quickly  if  the  denser  gas  were  placed 
a^  ^ne  ^°P>  being  assisted  by  gravity;  but  the 
significant  fact  is  that,  with  the  denser  gas 
below,  mixing  takes  place  although  opposed  by  gravity.  A 
like  mixing  of  gases  with  the  air  is  familiar  in  cases  where 
the  presence  of  the  gas  can  be  detected  by  its  odor,  as 
when  illuminating  gas  is  permitted  to  escape  from  a  burner 
or  a  little  ammonia  is  poured  on  the  floor. 

The  essential  facts  in  such  phenomena  are  these:  (i) 
The  gases  mix  without  outside  aid  (such  as  gravity,  stir- 
ring, or  currents  of  air).  The  mixing  is  spontaneous;  i.e. 
it  takes  place  from  internal  causes  —  causes  depending 


FIG  1  68 


MOLECULAR  PROPERTIES  OF  GASES 


203 


on  the  nature  of  gases.  (2)  The  final  result,  if  sufficient 
time  is  allowed,  is  always  a  uniform  mixture  of  the  gases, 
regardless  of  their  relative  densities.  (3)  The  less  the  den- 
sity of  a  gas  the  more  rapidly  does  it  mix  with  other  gases. 
(4)  Mixing  is  more  rapid  at  higher  temperatures.  (5)  All 
gases  and  vapors  behave  in  this  way. 

The  spontaneous  mixing  of  gases  is  called  diffusion. 
Diffusion  is  explainable  only  upon  the  supposition  that  the 
molecules  of  a  gas  are  in  constant 
motion  as  individuals,  darting 
hither  and  thither  in  all  direc- 
tions at  random,  like  bees  or 
gnats  in  a  swarm.  Each  molecule, 
according  to  this  view,  flies  in  a 
straight  line  till  it  hits  another 
molecule  or  the  wall  of  the  con- 
taining vessel,  when  it  rebounds 
like  a  perfectly  elastic  body. 
The  following  experiment  lends 
emphasis  to  these  conclusions. 


FIG.  169.  —  Diffusion  of  Gases. 


A  porous  cup  of  unglazed  earthen- 
ware is  fitted  with  a  stopper  and  a 
glass  tube,  and  supported  with  the 
lower  end  of  the  tube  in  a  glass  of 
water  (Fig.  169).  A  jar  or  a  large  beaker  is  held  over  the  cup 
and  filled  with  coal  gas  (or,  better,  hydrogen)  through  a  rubber 
tube.  The  experiment  has  three  stages,  (i)  Air  is  forced  out  through 
the  lower  end  of  the  tube,  showing  an  increase  of  pressure  inside  the 
porous  cup.  (2)  On  removing  the  jar,  water  rises  rapidly  in  the  tube, 
indicating  a  rapid  decrease  of  pressure  within  the  cup.  (3)  The 
water  in  the  tube  soon  begins  to  fall,  but  more  slowly  than  it  rose, 
and  finally  reaches  the  same  level  as  in  the  glass,  showing  that  at- 
mospheric pressure  has  been  restored.  These  phenomena  are  evi- 
dently due  to  the  rapid  diffusion  of  the  coal  gas  and  the  less  rapid 
diffusion  of  air  through  the  microscopic  pores  of  the  cup.  In  the  first 


204         THE   MOLECULAR  THEORY  OF  MATTER 

stage  of  the  experiment  the  coal  gas  diffuses  inward  more  rapidly 
than  the  air  inside  escapes;  hence  the  increase  of  pressure.  Jn  the 
second  stage  the  coal  gas  within  the  cup  escapes  more  rapidly  than 
air  enters,  causing  a  decrease  of  pressure.  Finally  air  diffuses  inward 
until  equilibrium  of  pressures  is  restored.  There  is  then  equally 
rapid  diffusion  of  air  inward  and  outward. 

Diffusion  is  a  very  different  process  from  the  flow  of 
gases  in  currents.  In  diffusion  the  molecules  move  singly 
and  at  random;  in  currents  they  move  collectively  as  one 
body.  Diffusion  supplements  the  action  of  winds  in  keep- 
ing the  constituents  of  the  air  uniformly  mixed. 

171.   Other  Phenomena  Explained  by  Molecular  Motion. 

-  The  pressure  of  a  gas  is  a  necessary  consequence  of  the 
motion  of  its  molecules.  The  walls  of  a  vessel  containing 
a  gas  are  subjected  to  an  inconceivably  rapid  shower  of 
blows  from  the  flying  molecules.  The  net  result  is  a  con- 
tinuous and  constant  pressure,  like  the  pressure  of  a  stream 
of  water  from  a  hose  when  directed  against  the  side  of  a 
building.  The  pressure  exists  throughout  the  mass  of 
gas,  for  the  molecules  in  the  interior  collide  with  one  an- 
other. Each  molecule,  knocking  vigorously  about  among 
its  fellows,  tends  to  drive  them  farther  away;  hence  the 
entire  mass  tends  to  expand  indefinitely.  If  a  given  mass 
of  gas  is  compressed  to  half  its  former  volume,  its  density 
is  doubled,  and  twice  as  many  molecules  strike  each  square 
centimeter  of  the  wall  of  the  vessel  every  second;  hence  the 
pressure  is  doubled.  Thus  molecular  motion  accounts  for 
Boyle's  law. 

From  the  known  density  and  pressure  of  a  gas,  it  is  pos- 
sible to  compute  the  average  velocity  of  its  molecules.  For 
air  this  velocity  turns  out  to  be  about  eighteen  miles  per 
minute,  or  half  the  velocity  of  the  swiftest  cannon  ball. 
If  the  molecules  of  what  we  call  still  air  were  moving  with 


MOLECULAR  PROPERTIES  OF  GASES  205 

their  actual  velocity,  but  all  in  the  same  direction,  the  re- 
sult would  be  a  wind  blowing  ten  times  as  fast  as  the  most 
violent  hurricane.  The  average  velocity  of  hydrogen 
molecules  is  about  sixty-eight  miles  per  minute,  —  a  veloc- 
ity sufficient  to  encircle  the  earth  in  six  hours.  To  the 
thoughtful  pupil  such  molecular  velocities  may  well  seem 
incredible;  but  any  difficulty  that  may  be  experienced  in 
picturing  them  to  the  mind  is  not  to  be  regarded  as  evi- 
dence that  they  do  not  exist.  It  is  demonstrated  from 
astronomical  observations  and  by  actual  experiment  that 
the  velocity  of  light  is  186,000  miles  per  second,  and  it 
would  be  unreasonable  to  doubt  the  fact  merely  because 
we  are  unable  to  think  it. 

It  is  evident  from  the  very  great  compressibility  of  gases 
that,  under  ordinary  pressures,  their  molecules  occupy 
only  an  exceedingly  small  portion  of  the  space  allotted 
to  them.  The  volume  of  a  gram  of  steam  under  a  pressure 
of  one  atmosphere  is  1661  ccm.,  which  is  1661  times  as  great 
as  its  volume  as  a  liquid.  A  cubic  inch  of  water  makes 
nearly  a  cubic  foot  of  steam,  and  that  without  any  increase 
in  the  number  or  (so  far  as  we  know)  in  the  size  of  the 
molecules.  Even  if  the  molecules  were  packed  without 
space  between  them  in  the  liquid  (which  is  not  true),  the 
average  distance  between  the  molecules  of  steam,  under  a 
pressure  of  one  atmosphere,  would  be  nearly  twelve  times 
the  diameter  of  a  molecule,  and  there  are  good  reasons 
for  believing  that  it  is  much  more  than  this.  Oxygen  has 
been  subjected  to  a  pressure  of  3000  atmospheres,  in  which 
condition  its  density  is  greater  than  that  of  water,  although 
it  still  remains  in  the  gaseous  state.  The  relatively  wide 
separation  of  the  molecules  of  a  gas  at  ordinary  tem- 
peratures and  pressures  is,  as  we  have  seen,  the  result  of 
molecular  motion.  Thus  molecular  motion  fully  accounts 


206         THE  MOLECULAR  THEORY  OF  MATTER 

for  the  great  compressibility  of  gases,  as  well  as  for  their 
great  resistance  to  compression. 

Considering  the  great  velocity  of  gas  molecules,  the  comparatively 
slow  rate  at  which  two  gases  mix  by  diffusion  requires  a  word  of 
explanation.  Diffusion  progresses  by  the  forward  motion  of  the 
molecules  of  each  gas  between  those  of  the  other.  But  molecules 
are  rather  numerous,  although  relatively  far  apart;  and  it  is  esti- 
mated that,  at  ordinary  temperatures  and  pressures,  each  collides 
with  its  neighbors  some  five  thousand  million  times  per  second. 
Hence  progression  of  the  molecules  in  any  given  direction  is  frequently 
interrupted  and  is  relatively  slow.  When  a  gas  is  admitted  into  a 
vacuum,  there  is  no  opposition  to  progressive  motion,  and  expan- 
sion is  practically  instantaneous. 

172.  Heat  and  Molecular  Motion.  —  When  a  gas  is 
heated  it  expands  or,  if  expansion  is  prevented,  its  pres- 
sure increases.  Some  idea  of  the  rate  of  expan- 
sion can  be  gained  by  means  of  a  small  flask 
containing  air,  and  having  a  glass  tube  attached 
(Fig.  170).  If  the  flask  is  inclosed  in  the 
hands  while  the  end  of  the  tube  is  held  under 
water,  the  expansion  will  be  shown  by  the 
escape  of  bubbles  from  the  tube.  If  the  flask 
is  then  allowed  to  cool  while  the  tube  remains 
under  water,  the  water  will  rise  in  it,  showing 
that  the  volume  and  the  pressure  of  the  air 
have  both  decreased.  These  effects  can  be 
due  only  to  an  increase  of  molecular  velocity 
''  I7°'  with  a  rise  of  temperature  and  a  decrease  of 
molecular  velocity  with  a  fall  of  temperature.  Thus,  in 
general,  when  a  gas  receives  heat  its  molecules  move  faster; 
when  it  loses  heat  they  move  more  slowly.  Since  we  know 
that  heat  is  a  form  of  energy,  we  might  suspect,  without 
further  evidence,  that  the  heat  of  a  gas  is  the  kine tic- 
energy  of  its  molecules;  and  such  is,  in  fact,  the  case. 


MOLECULAR   PROPERTIES  OF  LIQUIDS          207 

173.  The  Kinetic  Theory  of  Gases.  —  The  theory  upon 
which  we  have  relied  in  the  preceding  pages  to  account 
for  the  physical  properties  of  gases  is  known  as  the  kinetic 
theory  of  gases.     In  its  complete  form,  as  presented  in 
advanced  texts,  the  theory  applies  the  laws  of  dynamics 
to  the  individual  molecules,  and  accounts  definitely  for 
all  the  laws  of  gases.     There  are  many  points  of  minor 
importance  on  which  the  theory  is  either  silent  or  gives 
only  a  provisional  answer;  but  in  all  its  essential  features 
(and  we  have  considered  only  these)  it  is  regarded  by  sci- 
entists as  established  fact. 

III.   MOLECULAR  PROPERTIES  OF  LIQUIDS 

174.  Diffusion  of  Liquids.  —  If  any  two  liquids  that  can 
be  mixed  with  each  other  are  placed  in  the  same  vessel, 
the  denser  at  the  bottom,  and  left  undisturbed, 

they  will  mix  by  diffusion,  the  process  being  sim- 
ilar to  the  diffusion  of  gases.  The  progress  of 
diffusion  in  liquids  is  visible  in  cases  where  it  is 
accompanied  by  a  change  of  color,  as  in  the 
following  experiments. 


A  tall  vessel  (Fig.  171)  is  nearly  filled  with  water 
colored  with  blue  litmus.  A  little  strong  sulphuric  acid 
is  then  admitted  at  the  bottom  through  a  thistle-tube. 
The  acid  is  considerably  denser  than  the  water  and 
supports  it,,  the  surface  separating  the  two  being  dis- 
tinctly visible.  Since  acid  turns  blue  litmus  red,  the 
progressive  change  of  color  from  blue  to  red,  which  FIG.  171. 
slowly  takes  place  up  the  tube,  indicates  the  height  to  Diffusion 
which  the  acid  has  risen  by  diffusion. 

A  jar  is  partly  filled  with  water,  and  a  strong  solution  of  copper 
sulphate  is  admitted  at  the  bottom  through  a  thistle-tube.  The 
progress  of  diffusion  is  indicated  by  the  very  slow  rise  of  the  blue 
color  of  the  solution.  The  process  requires  months  for  its  completion. 


208         THE  MOLECULAR  THEORY  OF  MATTER 

We  see  from  these  experiments  that  diffusion  takes  place 
in  liquids,  as  in  gases,  without  the  aid  of  currents  and  in 
opposition  to  gravity.  The  explanation  is  therefore  the 
same:  the  molecules  of  a  liquid  are  in  individual  motion. 
The  exceedingly  slow  rate  of  diffusion  in  liquids  is  due  to 
the  frequent  collisions  of  the  molecules,  and  is  not  to  be 
taken  as  evidence  that  molecular  motion  is  sluggish.  The 
molecules  of  a  liquid  are  always  moving  about  among  one 
another  whether  a  second  liquid  is  present  or  not;  but  in 
the  latter  case  there  is  no  direct  evidence  of  the  motion. 

175.   Evidence  of  Intermolecular  Spaces  in  Liquids.  - 

Since  the  molecules  of  a  liquid  are  in  motion,  the  question 
naturally  arises  whether  they  are  moving  fast  enough  to 
drive  one  another  apart,  leaving  intermolecular  spaces,  or 
whether  their  mutual  attraction  (cohesion)  is  sufficient 
to  hold  them  actually  in  contact,  while  permitting  them 
to  slip  about  among  one  another.  Certain  properties  of 
liquids  supply  the  answer. 

Compressibility.  —  A  pressure  of  3000  atmospheres 
diminishes  the  volume  of  water  by  one  part  in  ten  and  the 
volume  of  sulphuric  ether  by  approximately  one  part  in 
six.  All  liquids  tested  under  great  pressure  have  been 
found  to  be  compressible  in  a  greater  or  less  degree.  The 
reasonable  inference  is  that  the  molecules  of  a  liquid  are 
separated  by  void  spaces,  and  that  in  compression  they  are 
simply  crowded  more  closely  together.  The  only  alterna- 
tive is  that  the  molecules  themselves  are  diminished  •  in 
size  by  compression;  but  this  view  is  discredited  by  other 
facts. 

Effect  of  Heat.  —  The  change  of  volume  of  a  liquid  when 
heated  or  cooled  can  be  shown  with  a  flask  and  tube  (Fig. 
170).  When  the  flask  is  completely  filled  with  cold  water 


MOLECULAR  PROPERTIES   OF  LIQUIDS  209 

or  other  liquid  and  plunged  into  a  vessel  of  hot  water,  the 
liquid  rises  rapidly  in  the  tube.  If  the  flask  is  now  plunged 
into  cold  water,  the  liquid  in  the  tube  descends.  When 
water  is  heated  from  the  freezing  to  the  boiling  point,  it 
increases  in  volume  by  four  per  cent.  The  expansion  of 
liquids  with  heat,  although  much  less  than  that  of  gases, 
is  plainly  due  to  the  same  cause,  —  heat  increases  molec- 
ular motion,  and  the  molecules  drive  one  another  farther 
apart. 

Loss  of  Volume  in  Mixing  Liquids.  — When  equal  volumes 
of  water  and  strong  alcohol  are  mixed,  it  is  found  that  the 
volume  of  the  mixture  is  two  per  cent  less  than  the  sum  of 
the  original  volumes.  This  is  readily  shown  by  filling  a 
long  test  tube  half  full  of  water  and  adding  alcohol  care- 
fully, to  avoid  mixing,  till  the  tube  is  nearly  full.  A  rub- 
ber band  round  the  tube  conveniently  marks  the  exact 
height.  The  tube  is  then  closed  with  the  finger,  and  the 
liquids  are  mixed  by  shaking.  A  similar  shrinkage  occurs 
in  mixing  strong  sulphuric  acid  and  water,  and  also  when 
certain  solids,  as  sugar  or  salt,  are  dissolved  in  water. 
There  is,  of  course,  no  loss  of  matter  in  such  phenomena; 
weight  and  mass  are  unchanged.  Evidently  the  mole- 
cules are  packed  together  more  closely  in  the  mixture 
than  in  the  separate  substances;  and  hence  there  must 
have  been  unoccupied  spaces  between  the  molecules  before 
the  mixing. 

Such  phenomena  as  these  lead  with  practical  certainty 
to  the  conclusion  that  there  are  intermolecular  spaces 
or  pores  in  all  liquids,  as  in  gases,  and  that  these  spaces 
are  maintained  by  decidedly  vigorous  molecular  motion. 
This  motion  is  principally  an  irregular  oscillation;  for  the 
molecules  are  so  close  together  that  nearly  all  their  time  is 
spent  in  collisions  with  one  another.  They  do,  however, 


210    THE  MOLECULAR  THEORY  OF  MATTER 

wander  uncertainly  about,  as  is  proved  by  diffusion.  Their 
mutual  attraction  (cohesion)  is  sufficiently  strong  to  set 
a  limit  to  expansion,  without  the  aid  of  external  pressure. 

176.  Surface  Tension  of  Liquids.  —  A  pin  or  a  needle 
that  is  slightly  oily  from  contact  with  the  fingers  can  be 
made  to  float  on  water,  if  carefully  laid  upon  the  surface. 
The  floating  pin  lies  in  a  depression  of  the  surface  which  is 
considerably  larger  than  itself  (represented  in  cross-sec- 
tion in  Fig.  172).  This  behavior  can  not  be  referred  to  the 
principle  of  buoyancy,  since  the  pin  is  much 
if§  denser  than  water.  If  the  pin  is  pushed 
O^jfMMJGf  beneath  the  surface  it  immediately  sinks. 
FIG.  172.  The  surface  of  the  water  behaves  as  if 
it  had  a  certain  degree  of  toughness  and  resisted  tearing. 

Soap  films,  whether  flat  or  in  the  form  of  bubbles,  exhibit 
this  property  in  unmistakable  fashion.  A  bubble  left 
upon  a  pipe  slowly  contracts,  driving  the  air  out  through 
the  stem.  It  behaves  just  as  an  inflated  rubber  balloon 
does  when  its  tube  is  opened.  The  film  of  the  bubble  and 
the  rubber  of  the  inflated  balloon  are  under  tension  and 
tend  to  contract.  Since  a  spherical  surface  is  smaller  than 
any  other  that  incloses  an  equal  volume,  both  the  bubble 
and  the  balloon  assume  this  shape  in  shrinking  as  much 
as  the  pressure  of  the  inclosed  air  or  other  gas  will  permit. 

A  drop  of  any  liquid  when  freed  from  the  distorting  effect 
of  its  weight,  as  in  falling,  is  spherical.  A  drop  of  oil 
suspended  in  a  solution  of  alcohol  and  water  of  its  own 
density  is  an  excellent  illustration.  The  spherical  form  of 
a  drop  is  due  neither  to  the  mutual  gravitation  of  its  par- 
ticles nor  to  cohesion  acting  throughout  the  mass,  but  to 
the  tension  of  its  surface.  It  is  as  if  the  drop  were  inclosed 
in  a  little  rubber  bag,  tightly  stretched. 


MOLECULAR  PROPERTIES  OF  LIQUIDS          21 1 

Every  liquid  may  be  regarded  as  bounded  by  a  skin  or 
film,  which  behaves  like  a  stretched  membrane.  The  ten- 
sion of  this  superficial  film  is  called  the  surface  tension  of 
the  liquid. 

The  surface  tension  of  different  liquids  has  been  deter- 
mined by  experiment,  and  has  been  found  to  be  greater 
for  water  than  for  any  other  liquid  except  mercury;  hence 
the  surface  tension  of  water  is  diminished  by  mixing  any 
other  substance  with  it.  This  is  readily  shown  by  placing 
a  drop  of  alcohol,  ether,  oil,  or  soap  solution  on  the  surface 
of  a  glass  of  water  beside  a  floating  sliver  of  wood,  as  a 
toothpick.  The  toothpick  is  quickly  jerked  away  from 
the  drop  by  the  greater  tension  of  the  pure  water  on  the 
other  side. 

177.  The  Cause  of  Surface  Tension.  —  Imagine  a  spheri- 
cal surface,  whose  radius  is  equal  to  the  range  of  cohesion, 
to  be  described  about  any  molecule  A 

(Fig.  173)  within  a  liquid.     (This  sphere 
is  microscopically  small;  the  figure  repre- 
sents   it    enormously   magnified.)     Since 
all  the  molecules  that  are  near   enough 
to  attract  A  are  uniformly  distributed 
about  it  within  this   sphere,   it  is  equally  attracted   in 
all  directions,  and  the  resultant  of  these  attractions  is 
zero.     But  any  molecule,  B  or  C,  whose  distance  from  the 
surface  is  less  than  the  range  of  cohesion,  is  more  strongly 
attracted  inward  than  outward,  since  most  of  the  mole- 
cules within   its    sphere  of   attraction   lie  on  the  inside. 
Upon  such  molecules  there  is  a  resultant  force  of  cohesion, 
acting  inward;  and  this  gives  rise  to  surface  tension. 

178.  Capillary  Action.  —  The  free  surface  of  a  liquid  is 
generally  sharply  curved  where  it  comes  in  contact  with  a 


212    THE  MOLECULAR  THEORY  OF  MATTER 

solid.  The  direction  of  the  curvature  depends  upon  the 
relative  strength  of  cohesion  in  the  liquid  and  adhesion 
between  the  liquid  and  the  solid.  If  adhesion  is 
the  greater,  the  edge  of  the  liquid  is  drawn  up- 
ward against  the  surface  of  the  solid.  Water  in 
contact  with  glass  is  a  familiar  example  (Fig.  174). 
If  cohesion  in  the  liquid  is  the  greater,  the  edge 
FIG.  1 74-—  of  the  liquid  is  drawn  away  from  the  solid,  and 

Contact  of  * 

Water  and  the   curvature   is   downward.     This   is    true   of 
1SS'      mercury  in  contact  with  glass  (Fig.  175). 
Surface  tension  plays   an  important  part  in  all  such 
phenomena.     When  a  liquid  creeps  up  over  the  surface 
of  a  solid,  as  from  b  to  c  (Fig.  176),  its  own 
surface  is  increased;   but  instead  of  turning  at 
an  angle,  abc,    it  contracts  into  the  curve  ab'c. 
This   curve  has  a  definite  shape,  in  which  the 
attraction  of  the  solid,  the  tension  of  the  liquid 
surface,  and  the  weight  of  the  raised  liquid  are  JIG'I75-~T 

u\  °  '  Contact  of 

in  equilibrium.     If  the  liquid  does  not  wet  the     Mercury 
solid,  its  surface  is  under  tension  even  where  a 
it  comes  in  contact  with  the  solid;  and,  in  contracting  as 
much  as  possible,  it  rounds  off  the  edge. 
These  and  other  effects  of  molecular  forces,  where  a 
liquid  comes  in  contact  with  a  solid,  are  known  as 
a  capillary  phenomena,  because  they  are  most  con- 
spicuous   in    tubes    of   small   or   hair-like   bore 
(Latin,  capillus,  a  hair).     Capillary  action  and 
capillarity  are  general  terms  for  such  phenomena 
FlG- I76'    or  for  their  cause. 

179.  Phenomena  in  Capillary  Tubes.  —  When  a  glass 
tube  of  small  bore  is  placed  with  its  lower  end  in  water, 
the  water  rises  in  the  tube  above  its  level  in  the  vessel, 


FIG.  177.  —  Capillary 
Elevation. 


MOLECULAR   PROPERTIES  OF  LIQUIDS          213 

and  comes  to  rest  with  its  surface  concave,  viewed  from 
above.  With  tubes  of  unequal  bore,  the  water  stands 
higher  in  the  smaller  and  its  surface 
is  more  sharply  curved.  The  ele- 
vation of  the  water  in  the  tube  is 
merely  an  exaggerated  instance  of 
capillary  action,  as  explained  above. 
The  water  is  drawn  up  at  its  edge 
by  adhesion  to  the  inner  wall  of  the 
tube.  The  surface  thus  becomes 
curved,  but  surface  tension  tends 
to  keep  it  flat  by  contraction;  hence 
the  entire  surface  rises.  At  a  certain 
height  the  weight  of  the  water  col- 
umn lifted  up  balances  the  upward 
force  due  to  adhesion  and  surface  tension.  As  the  size  of  the 
bore  is  reduced,  the  lifting  force  decreases,  but  the  weight 
of  water  per  unit  length  of  the  column 
decreases  still  more  rapidly;  hence  a 
higher  column  is  necessary  to  estab- 
lish equilibrium  (Fig.  177). 

Mercury  stands  in  small  glass 
tubes  below  the  level  of  its  surface 
in  the  vessel,  and  the  smaller  the 
bore  the  greater  the  depression  (Fig. 
178).  The  surface  is  convex,  for  rea- 
sons stated  above,  and  consequently 
exerts  a  downward  pressure. 

The  following  laws    have   been 
established  by  experiment: 
i.   If  a  liquid  wets  a  capillary  tube,  its  surface  is  concave 
and  it  is  drawn  up;  if  it  does  not  wet  the  tube,  Us  surface  is 
convex  and  it  is  depressed. 


FIG.  178.  —  Capillary 
Depression. 


214         THE   MOLECULAR  THEORY  OF   MATTER 

2.  The  elevation  or  the  depression  in  a  capillary  tube  is 
inversely  proportional  to  the  diameter  of  the  tube. 

Porous  solids  absorb  liquids  by  capillary  action  in  their 
minute  openings.  The  absorption  of  water  by  a  sponge 
or  a  towel,  of  ink  by  blotting  paper,  and  of  coffee  by  a  lump 
of  sugar  are  familiar  examples.  The  flame  of  a  lamp  is 
fed  by  oil  that  is  drawn  up  through  the  wick  by  capillary 
action.  In  dry  weather  moisture  is  drawn  up  from  a  depth 
of  many  feet  through  the  pores  of  the  soil,  and  evaporates 
at  the  surface.  Cultivation  of  the  soil  increases  the  size 
of  the  pores,  and  consequently  checks  the  rise  of  water 
through  the  cultivated  layer,  thus  diminishing  the  loss  by 
evaporation  at  the  surface. 

IV.    MOLECULAR  PROPERTIES  OF  SOLIDS 

180.  Intermolecular  Spaces  in  Solids.  —  Many  solids 
are  visibly  porous,  either  to  the  unaided  eye  or  with  the 
aid  of  a  microscope.  Paper,  wood,  leather,  brick,  and  sand- 
stone are  familiar  examples.  The  absorption  of  water  and 
other  liquids  by  solids  is  due  to  capillary  action  in  such 
pores  as  these.  But  the  metals  and  many  other  solids 
do  not  absorb  liquids,  and  appear  to  have  a  perfectly  con- 
tinuous structure  even  under  the  microscope;  nevertheless, 
there  is  direct  experimental  evidence  that  such  bodies 
have  invisible  openings  in  and  through  them.  Early  in 
the  seventeenth  century,  Francis  Bacon,  experimenting  on 
the  compressibility  of  water,  hammered  a  shell  of  lead 
rilled  with  water.  "  The  water  exuded  like  perspiration 
through  the  pores  of  the  lead.  The  Florentine  Academi- 
cians tried  the  same  experiment  with  a  silver  shell,  but 
obtained  the  same  result.  They  then  tried  to  prevent 
the  escape  of  the  water  by  thickly  gilding  the  shell,  but 
again  in  vain."  Under  a  pressure  of  4000  atmospheres, 


MOLECULAR  PROPERTIES  OF  SOLIDS  215 

mercury  has  been  forced  through  three  inches  of  solid  steel. 
In  such  cases  as  these  it  is  highly  probable  that  the  pores 
are  nothing  more  than  spaces  between  the  individual 
molecules  of  the  solid. 

Further  evidence  of  intermolecular  spaces  in  solids  is 
afforded  by  their  change  of  volume  with  dhange  of  pres- 
sure or  of  temperature.  All  solids  are  compressible,  al- 
though many  are  less  so  than  liquids.  The  rolling  and 
stamping  to  which  silver  is  subjected  in  the  process  of 
coining  causes  a  decrease  of  volume  amounting  to  about 
four  per  cent.  Glass  is  about  one- twentieth  as  compress- 
ible as  water.  Nearly  all  solids  expand  with  heat.  The 
unequal  expansion  of  a  glass  dish,  when  hot  water  is  poured 
into  it,  strains  it  to  the  breaking  point.  The  rails  of  a 
track  are  laid  with  a  space  between  their  ends  to  provide 
room  for  expansion  in  hot  weather.  Expansion  is  readily 
demonstrated  with  a  small  brass  ring  and  u  brass  ball  that 
will  just  pass  through  it  when  both  are  cold  or  both  hot. 
If  only  the  ball  is  heated  it  will  not  go  through  the  ring. 

181.  Molecular  Motion  in  Solids.  —  Intermolecular 
spaces  in  solids  can  only  be  due  to  molecular  motion,  as 
in  liquids  and  gases.  Obviously  the  molecules  of  a  solid 
are  not  free  to  wander  about  among  one  another;  but  we 
can  imagine  an  irregular  oscillation  without  change  of 
position.  This  motion  must  be  very  energetic  indeed  to 
enable  a  solid  to  resist  compression  as  it  does,  or  to  cause 
the  observed  expansion  of  a  solid  when  heated.  Under 
ordinary  conditions  the  tendency  of  solids  to  expand  with 
a  rise  of  temperature  is  practically  irresistible.  To  stretch 
a  steel  rod  having  a  cross-section  of  one  square  inch  as  much 
as  it  expands  of  itself  when  it  is  warmed  from  the  tempera- 
ture of  melting  ice  to  the  ordinary  temperature  of  a  room 
(20°  Centigrade)  would  require  a  tension  of  8000  lb.;  and 


2i6         THE   MOLECULAR  THEORY  OF   MATTER 

it  would  take  an  equal  pressure  to  prevent  the  expansion 
of  the  rod  with  this  rise  of  temperature. 

The  universally  accepted  view  that  heat  is  the  energy 
of  molecular  motion  necessarily  implies  that  the  molecules 
of  all  bodies  —  solid,  liquid,  and  gaseous  —  are  in  motion; 
for  we  know  that  there  is  still  heat  in  bodies  at  the  lowest 
temperatures  yet  attained.  But  the  consideration  of  such 
matters  must  be  left  to  the  next  chapter. 

182.  Special  Properties  of  Solids.  —  Solids  differ  widely  among 
themselves  in  many  of  their  physical  properties;  indeed,  it  is  by 
means  of  such  differences  that  we  distinguish  substances  from  one 
another.  Thus  one  substance  is  hard,  another  soft;  one  is  brittle, 
another  tough,  etc.  Special  properties  may  be  classed  as  mechanical, 
optical,  magnetic,  electrical,  etc.  It  is  only  with  the  more  important 
mechanical  properties  that  we  are  now  concerned. 

An  elastic  solid  offers  permanent  resistance  to  change  of  shape; 
an  inelastic  or  plastic  solid  does  not.  Elasticity  of  form  has  been 
considered  at  length  in  an  earlier  chapter.  In  the  customary  and 
narrow  sense,  a  plastic  substance  is  one  that  can  be  molded  by  a 
moderate  pressure  into  any  desired  shape.  Putty  and  wet  clay  are 
typical  examples.  In  a  broader  sense,  all  solids  that  are  not  brittle 
are  plastic  beyond  their  elastic  limit.  Gold,  silver,  and  copper  are 
plastic  at  ordinary  temperatures,  when  subjected  to  great  pressures 
as  in  stamping  coins.  A  force  of  160  tons  is  applied  in  stamping  a 
silver  dollar;  and  under  this  force  the  cold  metal  behaves  like  butter 
in  a  butter  mold.  In  some  cases  even  brittle  substances  are  plastic 
under  the  continued  action  of  a  moderate  force.  A  stick  of  sealing 
wax  is  very  brittle;  but  when  supported  at  one  end  in  a  horizontal 
position,  it  slowly  yields  under  the  action  of  its  own  weight,  and  in 
the  course  of  weeks  becomes  quite  bent.  The  dividing  line  between 
plastic  solids  and  viscous  liquids  is  uncertain.  Various  substances, 
e.g.  molasses  candy,  pitch,  and  shoemakers'  wax,  change  imperceptibly 
to  the  liquid  state  by  gradual  softening,  when  heated. 

A  substance  is  said  to  be  malleable  if  it  can  be  hammered  or  rolled 
into  sheets,  ductile  if  it  can  be  drawn  out  into  a  wire.  Ordinary 
temperatures  are  understood  unless  otherwise  stated.  Thus  we  say 
that  glass  is  brittle,  although  it  is  very  ductile  when  heated  to  redness. 


MOLECULAR  PROPERTIES  OF  SOLIDS  217 

/ 

Gold,  silver,  copper,  and  platinum  are  the  most  ductile  metals,  gold 
the  most  malleable.  Gold  can  be  reduced  to  sheets  having  a  thickness 
of  rauforo  of  an  inch. 

Our  idea  of  what  constitutes  a  hard  body  varies  with  the  substance. 
It  is  easier  to  cut  hard  butter  than  soft  wood;  and  we  speak  of  soft 
iron  and  a  hard  pillow,  although  we  should  doubtless  prefer  the 
latter  to  sleep  on.  The  meaning  of  these  terms  is  variable  in  another 
respect.  They  are  applied  to  quite  different  physical  properties. 
Thus  we  say  that  soapstone  or  talc  is  soft,  meaning  that  it  is  easily 
scratched;  and  also  that  rubber  is  soft,  meaning  that  it  offers  little 
resistance  to  change  of  shape.  Hardness  may  mean  the  opposite 
of  softness  in  either  of  these  senses.  Where  some  degree  of  exactness 
is  necessary,  hardness  is  expressed  in  terms  of  certain  standards. 
Thus  the  mineralogist's  scale  of  hardness  consists  of  a  series  of  ten 
minerals,  beginning  with  talc,  which  can  be  scratched  with  the 
finger  nail,  and  ending  with  the  diamond,  the  hardest  of  all  substances. 
Hardness,  referred  to  this  scale,  means  resistance  to  scratching. 

A  brittle  or  friable  substance  is  one  that  is  easily  broken  into  frag- 
ments by  a  blow,  e.g.  glass  and  coal.  Brittleness  and  toughness  are 
opposite  properties. 

The  tenacity  or  tensile  strength  of  a  substance  depends  upon  the 
cohesion  of  its  molecules.  It  is  one  of  the  most  important  properties 
of  building  materials,  and  its  measure  is  the  force  per  unit  area  neces- 
sary to  pull  the  body  apart.  Steel  is  the  most  tenacious  of  all  sub- 
stances, a  tension  of  180  Ib.  being  required  to  break  a  steel  wire  hav- 
ing a  cross-section  of  i  sq.  mm.  The  breaking  strength  of  a  copper 
wire  of  the  same  size  is  about  70  Ib.,  of  lead  wire  5  Ib.,  of  glass  14  Ib., 
and  of  oak  (in  the  direction  of  its  fibers)  15  Ib. 

PROBLEMS 

1.  According  to  the  molecular  theory,  how  does  heat  convert  a  solid  into 
a  liquid?   a  liquid  into  a  gas? 

2.  Why  is  ground  damp  under  a  board  or  a  stone  when  it  is  dry  all 
around? 

3.  The  bristles  of  a  paint  brush  or  the  hairs  of  one's  head  cling  together 
when  wet  in  air;  but  under  water  they  spread  apart  as  they  do  when  dry. 
Explain. 

4.  A  tray  made  of  wire  gauze  will  hold  water  to  a  slight  depth,  if  it  is 
poured  in  very  gently.     Explain. 


CHAPTER  VIII 
HEAT 

I.  NATURE  OF  HEAT 

183.  The  Caloric  Theory.  —  From  the  days  of  the  early 
Greek  philosophers  until  about  the  middle  of  the  nineteenth 
century,  heat  was  very  generally  believed  to  be  a  substance. 
According  to  the  accepted  view  of  the  eighteenth  century, 
this  supposed  substance,  then  known  as  caloric,  was  an 
invisible,    elastic   fluid,   without   weight,   whose   particles 
repelled  one  another  but  were  attracted  more  or  less  by 
the  ordinary  kinds  of  matter.     With  the  overthrow  of  this 
theory,  the  word  caloric  has  become  obsolete;  but  several 
of  its  derivatives  continue  in  use,  e.g.  the  calorie  is  a  unit 
quantity  of  heat,  and  the  vessel  in  which  substances  are 
placed  in  measuring  their  gain  or  loss  of  heat  is  called  a 
calorimeter. 

184.  Historical  Experiments  on  the  Nature  of  Heat.  — 

As  early  as  the  seventeenth  century  some  of  the  ablest 
scientists,  including  such  men  as  Boyle,  Francis  Bacon, 
Hooke,  and  Newton,  believed  heat  to  be  molecular  motion; 
but  nearly  two  hundred  years  elapsed  before  this  view 
again  came  into  prominence,  through  the  experiments  of 
Benjamin  Thompson  (1753-1814).  Although  an  Amer- 
ican, born  in  Massachusetts,  Thompson  is  better  known 
as  Count  Rumford,  having  received  that  title  from  the 
Elector  of  Bavaria,  whose  service  he  entered  while  still  a 

218 


NATURE  OF  HEAT  219 

young  man.  While  engaged  in  boring  cannon  for  the 
Bavarian  government,  he  was  surprised  at  the  heat  gen- 
erated, and,  seeking  further  light  on  the  phenomenon,  he 
began  to  experiment.  "He  arranged  apparatus  so  that 
the  heat  generated  by  the  friction  of  a  blunt  steel  borer 
raised  the  temperature  of  a  quantity  of  water.  In  his 
third  experiment,  water  rose  in  one  hour  to  107°  Fahrenheit; 
in  one  hour  and  a  half  to  142°;  at  the  end  of  two  hours  and 
a  half  the  water  actually  boiled.  '  It  is  difficult  to  describe 
the  surprise  and  astonishment,'  says  Rumford,  ' expressed 
in  the  countenances  of  the  bystanders,  on  seeing  so  large 
a  quantity  of  cold  water  (18.75  lb.)  heated,  and  actually 
made  to  boil  without  any  fire.'  The  source  of  heat  gener- 
ated by  friction  '  appeared  evidently  to  be  inexhaustible.' 
The  reasoning  by  which  he  concluded  that  heat  is  not  mat- 
ter, but  is  due  to  motion,  we  can  give  only  in  part.  He 
says,  '  It  is  hardly  necessary  to  add  that  anything  which 
any  insulated  body,  or  system  of  bodies,  can  continue  to 
furnish  without  limitation  cannot  possibly  be  a  material 
substance;  and  it  appears  to  me  extremely  difficult,  if  not 
quite  impossible,  to  form  any  distinct  idea  of  anything 
capable  of  being  excited  and  communicated  in  the  manner 
in  which  heat  was  'excited  and  communicated  in  these 
experiments,  except  it  be  motion.' 

"Rumford's  conclusion  regarding  the  nature  of  heat  was 
vigorously  attacked  by  the  calorists,  but  it  was  confirmed 
in  1799  by  Sir  Humphry  Davy.  By  means  of  clockwork 
he  rubbed  two  pieces  of  ice  against  one  another  in  the  vac- 
uum of  an  air  pump.  Part  of  the  ice  was  melted,  although 
the  temperature  of  the  receiver  was  kept  below  the  freez- 
ing point.  From  this  he  concluded  that  friction  causes 
vibration  of  the  corpuscles  of  bodies,  and  this  vibration 
is  heat."  (Cajori's  History  of  Physics.) 


220  HEAT 

The  views  of  Rumford  and  Davy  were  accepted  by  scientists  here 
and  there;  but  the  caloric  theory  was  too  firmly  established  to  be 
easily  supplanted  by  its  rival.  The  prevailing  opinion  continued  in 
its  favor  until  the  middle  of  the  nineteenth  century;  at  which  time 
the  equivalence  of  the  different  forms  of  energy,  including  heat,  and 
the  principle  of  the  conservation  of  energy  were  established  by  the 
experiments  of  J.  P.  Joule  in  England  (Art.  242)  and  the  discussions 
of  Robert  Mayer  and  von  Helmholtz  in  Germany. 

185.  The  Kinetic  Theory  of  Heat.  —The  molecular  prop- 
erties of  matter  (Chap,  vii)  afford  conclusive  evidence  that 
the  molecules  of  all  bodies  —  solid,  liquid,  and  gaseous  — 
are  in  motion,  and  that  this  motion  is  increased  by  heat. 
The  production  of  heat  from  mechanical  energy  proves 
that  heat  is  itself  a  form  of  energy,  and  that  this  energy  is 
associated  with  the  molecules  of  bodies.     Since  a  moving 
molecule  must  possess  kinetic  energy  by  virtue  of  its  mass 
and  its  velocity,  as  in  the  case  of  large  masses,  it  follows 
that  this  kinetic  energy  of  molecular  motion  must  be  heat. 
"  Heat  is  not  motion,  for  it  is  neither  change  of  position, 
nor  yet  momentum;  it  is  the  energy  of  motion.     Double 
the  quantity  of  molecular  motion,  and  you  quadruple  the 
molecular  kinetic  energy,  that  is,  the  heat."     (Daniell.) 

"Cold,"  in  the  sense  of  something  whose  effects  are 
opposite  to  those  of  heat,  does  not  exist.  Ice  does  not 
"give  out  cold";  it  has  none  to  give  out.  It  cools  sur- 
rounding objects  by  receiving  heat  from  them ;  and  because 
it  receives  heat  it  melts.  Some  heat  remains  in  all  bodies 
at  the  lowest  temperatures  yet  produced. 

II.   TEMPERATURE 

* 

186.  Temperature.  —  Our    first    ideas    of    temperature 
are  derived  from  our  bodily  sensations  of  heat  and  cold; 
and  these  ideas  are  expressed  in  such  terms  as  hot,  warm, 


TEMPERATURE  221 

tepid,  cool,  cold,  etc.  The  warmer  of  two  bodies  is  said 
to  be  at  the  higher  temperature.  The  terms  higher  and 
lower  suggest  the  possibility  of  exact  measurement;  and 
the  use  of  the  thermometer  for  this  purpose  is  familiar  to 
every  one. 

When  any  two  bodies  whose  temperatures  differ  are 
placed  in  contact,  heat  passes  of  itself  from  the  hotter  to 
the  colder,  until  they  reach  the  same  temperature.  The 
behavior  of  heat  in  passing  from  a  body  at  higher  to  one  at 
lower  temperature  is  similar  to  the  behavior  of  water  in 
flowing  from  higher  to  lower  level.  Temperature  may 
therefore  be  defined  as  that  condition  of  a  body  on  which  its 
ability  to  impart  heat  to  other  bodies,  or  to  receive  heat  from 
them,  depends.  It  should  be  noted  that  two  bodies  at 
equal  temperatures  may  or  may  not  contain  equal  quan- 
tities of  heat;  just  as  two  communicating  vessels  in  which 
water  stands  at  the  same  level  may  or  may  not  contain 
equal  quantities  of  water.  The  quantity  of  heat  in  a  body 
at  a  given  temperature  varies  as  its  mass,  and  depends 
also,  as  we  shall  see  later,  upon  the  material  of  the  body. 

Our  temperature  sensations  are  often  very  misleading.  In  the 
first  place,  the  sensation  varies  with  the  condition  of  our  own  bodies. 
A  room  may  seem  agreeably  warm  to  a  person  entering  it  after  a 
brisk  walk  on  a  cold  day,  while  to  another,  who  has  not  been  exercising, 
it  seems  chilly.  A  cold  room  seems  warm  to  one  who  is  ill  with  a  fever, 
and  a  warm  room  cold  to  one  who  has  the  "grippe."  In  the  second 
place,  the  temperature  sensation  experienced  on  touching  a  body  is 
largely  determined  by  certain  physical  properties  of  the  body  itself. 
If  there  has  been  no  fire  or  other  source  of  new  heat  in  a  room  for  an 
hour  or  more,  a  thermometer  would  show  the  various  objects  in  it  to 
be  at  the  same  temperature;  but  to  the  touch  they  do  not  seem  to  be 
so.  On  a  cold  morning  we  find  a  rug  or  a  carpet  fairly  comfortable 
to  the  bare  feet,  while  the  floor  feels  decidedly  cold.  A  wash-bowl 
feels  much  colder  to  the  hand  than  the  air  does,  and  the  water  in  it 
still  colder. 


222  HEAT 

187.  Measurement  of  Temperature.  —  Any  instrument 
for  measuring  temperature  is  called  a  thermometer.     Ther- 
mometers are  of  many  kinds,  and  make  use  of  various  effects 
of  heat,  such  as  the  expansion  of  a  solid,  a  liquid,  or  a  gas, 
the  change  of  pressure  of  a  gas,  and  the  change  of  electrical 
resistance  of  a  wire.     A  thermometer  in  which  the  expan- 
sion of  a  liquid  is  utilized  is  most  convenient  for  general 
use.     It  is  important  that  the  liquid  chosen  should  have  a 
uniform  expansion;  i.e.  equal  quantities  of  heat  should  cause 
equal  increases  of  volume  at  all  temperatures. 

Mercury  fulfils  this  condition  better  than  any  other 
liquid,  and  has  the  further  advantage  of  remaining  a  liquid 
through  a  very  wide  range  of  temperature.  The  mercury- 
in-glass  thermometer  has  therefore  been  adopted  as  the 
standard  for  all  ordinary  purposes.  For  temperatures 
below  the  freezing  point  of  mercury,  the  alcohol  thermom- 
eter is  generally  used,  the  freezing  point  of  alcohol  being 
—  130°  C.  The  hydrogen  thermometer  is  the  standard  for 
accurate  scientific  work.  In  it  a  mass  of  hydrogen  is  kept 
at  constant  volume,  and  the  pressure  of  the  hydrogen 
measures  the  temperature. 

188.  The  Mercury  Thermometer  has  a  capillary  glass 
tube,  called  the  stem,  terminating  in  a  bulb  (Fig.   180). 
The  mercury  fills  the  bulb  and  more  or  less  of  the  stem, 
according  to  the  temperature;  and  its  expansion  or  contrac- 
tion is  measured  by  a  scale  engraved  on  the  stem  or  at- 
tached to  it.     In  making  a  thermometer  the  mercury  is 
heated,  to  drive  out  all  the  air,  before  the  stem  is  sealed  at 
the  top;  hence  the  space  in  the  tube  above  the  mercury  is 
a  vacuum. 

Determination  of  the  Fixed  Points.  —  Probably  no  two 
thermometers  have  bulbs  of  exactly  the  same  capacity 


TEMPERATURE 


223 


and  tubes  of  exactly  the  same  bore;  hence  the  readings  of  dif- 
ferent thermometers  would  be  entirely  inconsistent  with  each 
other  if  they  were  provided  with  scales  of  the  same  length. 
The  correct  position  and  dimensions  of  the  scale  must  there- 
fore be  determined  separately  for  every  thermometer.  The 
first  step  in  this  process  is  to  determine  the  two  fixed  points, 
called  the  freezing  point  and  the  boiling  point. 

The  freezing  point  is  the  temperature  at  which  pure 
water  freezes;  but  since  this  is  exactly  the  same  as  the 
temperature  at  which  ice  melts,  whatever  the  surrounding 
temperature  may  be,  it  is  most  conveniently  found  by 
inserting  the  bulb  of  the  thermometer  in  a  dish  of  melting 
snow  or  ice.  The  snow  or  crushed  ice  is  packed  about 
the  bulb  and  stem,  leaving  the  mercury  just  visible  above 
it;  and  a  mark  is  made  on  the  stem  at  the  top  of  the 
mercury  column,  after  it  comes  to  rest. 

The  boiling  point  is  the  temperature  of  steam  as  it  rises 
from  water  boiling  under  a  pressure  of  one  atmosphere. 
The  temperature  of  boiling 
water  is  subject  to  slight  vari- 
ations from  different  causes; 
but  the  temperature  of  the 
steam  varies  only  with  the 
pressure.  The  thermometer 
is  therefore  adjusted  so  as  to 
be  surrounded  by  the  steam, 
as  nearly  as  possible  to  the 
top  of  the  mercury  in  the 
stem,  and  is  not  permitted  to 
touch  the  water  (Fig.  179). 
The  height  at  which  the  mer- 


FIG.  179. 


cury  stands,  under  these  conditions,  is  marked  on  the  stem 
as  the  boiling  point.     A  correction  must  be  applied  to  the 


224  HEAT 

observed  height  of  the  mercury  in  the  stem  if  the  pressure 

of  the  steam  is  not  76  cm.  (Art.  232). 

Centigrade  and  Fahrenheit  Scales. — The  distance  between 

the  fixed  points  is  divided  into  equal  parts,  called  degrees. 
In  the  Centigrade  scale  the  number  of  these 
divisions  is  100,  the  freezing  point  being  marked 
212.  o°  and  the  boiling  point  100°.  The  Centigrade 
thermometer  is  used  almost  exclusively  in  scien- 
tific work.  All  temperatures  referred  to  in  this 
book  are  expressed  in  the  Centigrade  scale,  unless 
otherwise  indicated.  In  the  Fahrenheit  scale  the 
freezing  point  is  marked  32°  and  the  boiling 
point  212°,  the  interval  between  them  being 
1 80°.  The  Fahrenheit  thermometer  is  the  one 

17.8  -  o°  in  general  use  in  English-speaking  countries. 
The  scale  of  a  thermometer  may  be  extended 
to  any  desired  distance  beyond  the  fixed  points. 
Temperatures  below  zero,  on  either  scale,  are 

FIG.  180.       .    j.  11,1  ,  • 

indicated  by  the  negative  sign. 

Since  the  interval  between  the  freezing  and  the  boiling 
points  is  100  Centigrade  degrees  or  180  Fahrenheit  degrees, 
it  follows  that  - 

i  Centigrade  degree  =  f  Fahrenheit  degree,  and 
i  Fahrenheit  degree  =  f  Centigrade  degree. 
In  changing  a  reading  on  either  scale  to  the  equivalent 
reading  on  the  other,  allowance  must  be  made  for  the  dif- 
ference in  the  zero  points.     Example:  50°  C.  means  50 
Centigrade   degrees   above    the    freezing   point.     This   is 
equal  to  50  X  I  or  90  Fahrenheit  degrees  above  the  freezing 
point,  or  to  122°  Fahrenheit. 

PROBLEMS 

1.  (a)  According  to  the  theory  of  heat,  what  would  be  the  molecular 
condition  of  a  body  without  heat?  (6)  Why  could  not  such  a  body  be  a  gas? 


CONDUCTION  AND  CONVECTION       225 

2.  Mention  any  familiar  instances  in  which  equal  temperatures  do  not 
cause  like  temperature  sensations. 

3.  (a)   The  reading  of  a  thermometer  gives  the  temperature  of  the 
thermometer.     On  what  grounds  do  we  assume  that  the  reading  of  a  ther- 
mometer in  a  liquid  gives  the  temperature  of  the  liquid?     (b)  Why  do  we 
not  take  the  reading  immediately  on  inserting  a  thermometer  in  a  liquid  to 
determine  its  temperature? 

4.  A  living  room  is  comfortable  at  a  temperature  of  67°  F.     What  is  this 
temperature  on  the  Centigrade  scale? 

5.  What     would    the     Centigrade    thermometer     register    in    "zero 
weather  "? 

6.  What  is  ''98  in  the  shade"  according  to  the  Centigrade  thermom- 
eter? 

III.   CONDUCTION  AND  CONVECTION 

189.  Conduction  is  the  transmission  of  heat  from  hotter 
to  colder  parts  of  a  body,  or  from  a  hotter  to  a  colder  body 
in  contact  with  it,  without  change  in  the  relative  positions 
of  the  parts  of  the  body.  It  is  the  only  process  by  which 
heat  travels  in  solids.  The  heating  of  the  farther  end  of 
a  poker,  when  one  end  is  placed  in  a  fire,  and  the  heating 
of  the  handle  of  a  spoon  placed  in  a  cup  of  hot  tea  are 
familiar  examples. 

The  kinetic  theory  suggests  a  mental  picture  of  the 
process  of  heat  conduction.  When  any  part  of  a  body  is 
heated,  its  molecules  are  set  in  more  rapid  vibration.  These 
molecules  jostle  their  neighbors  more  violently,  increasing 
the  energy  of  their  vibration.  The  disturbance  thus 
spreads  throughout  the  body  without  change  in  the  rela- 
tive positions  of  the  molecules  themselves.  In  conduction, 
therefore,  molecular  energy  is  transmitted  without  the 
transmission  of  matter. 

The  power  of  a  substance  to  transmit  heat  by  conduction 
is  called  its  conductivity.  This  property  varies  greatly 
with  different  substances.  Thus  a  burning  match  can  be 


226 


HEAT 


held  until  the  flame  reaches  the  fingers,  for  wood  is  a  poor 
conductor;  but  if  one  end  of  a  pin  is  held  in  the  flame  of 
the  match,  the  other  end  quickly  becomes  too  hot  to  hold. 
The  metals  are  the  best  conductors,  although  differing 
greatly  among  themselves;  and  other  solids,  with  few  excep- 
tions, are  better  conductors  than  liquids.  Liquids,  with 
the  exception  of  mercury  and  molten  metals,  are  poor 
conductors.  Water  can  be  boiled  at  the  top  of  a  test 

tube  for  several  minutes,  while 
at  the  bottom  it  remains  cold 
(Fig.  181).  But  a  number  of 
solids,  e.g.  wood,  paper,  and 
wool,  are  poorer  conductors 
than  water  (see  table  below). 
Gases  are  practically  non-con- 
ductors. In  testing  the  con- 
ductivity of  liquids  and  gases, 
they  must  be  heated  at  the 
top  to  prevent  convection  currents  (Art.  191). 

The  following  table  gives  the  conductivities  of  various 
substances  referred  to  silver  as  the  standard. 

TABLE  OF  CONDUCTIVITIES  FOR  HEAT   (APPROXIMATIONS) 


FIG.  181. 


Silver    100. 

Copper 80. 

Brass 27. 

Iron 15. 


Ice    5 

Marble 4 

Glass    15 

Water 14 


Mercury     1.6      Wood 04 


Writing  paper  ... .  .012 

Fresh  snow   01 

Felt 009 

Air   005 

Flannel 004 


190.  Temperature  Sensations  and  Conductivity.  Other 
Applications.  —  We  can  now  understand  in  part  why 
different  bodies,  e.g.  iron  and  wood,  feel  unequally  hot, 
when  actually  at  the  same  temperature  (Art.  186).  The 
difference  in  the  sensations  is  largely  due  to  the  unequal 
conductivities  of  the  substances;  for  the  sensation  is 


CONDUCTION  AND   CONVECTION  227 

determined  by  the  rate  at  which  the  hand  receives  or  loses 
heat.  Equally  hot  bodies,  differing  in  conductivity,  im- 
part heat  to  the  hand  at  unequal  rates,  and  thus  make 
the  hand  unequally  hot;  equally  cold  bodies  take  heat 
from  the  hand  at  unequal  rates,  making  the  hand 
unequally  cold.  The  unequal  temperatures  of  the  hand 
are  commonly  but  mistakenly  attributed  to  the  bodies 
themselves.  Temperature  sensations  are  also  determined 
in  part  by  another  property  of  bodies,  called  specific  heat, 
which  will  be  studied  later. 

Materials  of  low  conducting  power  are  widely  used  both  to  keep 
heat  from  cold  bodies  and  to  prevent  the  loss  of  heat  from  hot  bodies. 
The  double  walls  of  refrigerators  and  ice-houses 
are  rilled  in  between  with  charcoal,  sawdust, 
straw,  or  other  loose,  badly  conducting  mate- 
rial, to  keep  the  heat  out.  A  cooking  box 
(Fig.  182)  is  packed  with  felt  to  keep  the  heat 
in.  Food  placed  in  such  a  box  boiling  hot  will 
continue  to  cook  for  hours,  with  only  a  slight 
fall  of  temperature  and  without  a  further  appli- 
cation of  heat.  The  loss  of  heat  from  steam 
and  hot-air  pipes  is  greatly  reduced  by  a  FlG.  l82._CookingBox. 
wrapping  of  asbestos,  paper,  or  felt. 

The  low  conductivity  of  sawdust,  straw,  felt,  fur,  feathers,  hair, 
and  other  poor  conductors  is  largely  due  to  the  air  spaces  within 
them.  Air  is  one  of  the  poorest  conductors;  but  to  be  effective  it 
must  be  trapped  in  minute  cells,  which  prevent  its  circulation.  When 
not  thus  confined,  it  transfers  heat  by  convection  (Art.  191). 

191.  Convection.  —  We  have  seen  that  water  can  be 
heated  at  the  top  in  a  test  tube  and  boiled  while  it  remains 
cold  at  the  bottom.  If  the  flame  is  applied  at  the  bottom 
of  the  tube,  the  water  quickly  becomes  heated  throughout. 
This  is  evidently  not  the  result  of  conduction;  for  conduc- 
tion is  not  affected  by  gravity,  and  takes  place  upward 
and  downward  with  equal  rapidity.  When  a  portion  of  a 


228  HEAT 

liquid  is  heated,  it  expands  and  becomes  less  dense  than  the 
remainder.  If  this  heated  portion  is  at  the  bottom  of  the 
vessel,  it  will  be  driven  upward  by  the  cooler  and  denser 
portions  surrounding  it,  in  agreement  with  the  principle 
of  buoyancy.  As  the  colder  liquid  from  above  reaches 
the  place  where  the  heat  is  applied,  it  in  turn  becomes 
heated  and  is  displaced  by  other  portions.  The  process 
is  called  convection,  and  the  streams  of  warmer  and  cooler 
liquid  are  called  convection  currents.  When  the  heat  is 
applied  at  the  top,  the  expanded  liquid  is  already  in  the 
position  of  equilibrium  and  it  remains  there. 

Convection  takes  place  in  air  and  other  gases  under  the  same 
conditions  as  in  liquids.  The  strong  ascending  current  above  a  bon- 
fire is  indicated  by  the  leaping  of  the  flames  and  the  rapid  rise  of 
sparks  and  smoke.  The  fire  is  fed  by  inward-flowing  currents  near 
the  ground.  They  occupy  much  more  space  than  the  ascending 
current,  and  hence  move  more  slowly  and  are  less  noticeable. 

Convection  currents  can  be  made  visible  in  a  beaker  of  water  by 
means  of  sawdust  or  a  little  coloring  matter  (crystals  of  potassium 
permanganate)  placed  in  the  bottom  of  the  beaker  before  the  heat  is 
applied.  Convection  currents  in  air  are  made  visible  by  smoke, 
produced  by  burning  filter  paper  which  has  been  soaked  in  a  solution 
of  saltpeter  and  dried. 

192.  Applications  of  Convection.  —  The  draft  of  a  chimney  is  a 
convection  current.  It  is  maintained  by  an  upward  force  equal  to 
the  difference  between  the  weight  of  the  heated  air  in  the  chimney 
and  the  weight  of  a  column  of  outside  air  of  the  same  dimensions. 
(Why?)  Hence  a  tall  chimney  has  a  stronger  draft  than  a  low  one. 
A  lamp  chimney  maintains  a  steady  draft  of  air,  which,  entering  be- 
low, supplies  the  flame  with  the  oxygen  necessary  for  combustion. 
If  the  chimney  is  closed  at  top  or  bottom,  the  flame  at  once  begins 
to  smoke  from  imperfect  combustion,  and  is  quickly  extinguished. 

Convection  is  an  essential  process  in  the  heating  and  ventilation 
of  houses.  A  hot  stove  or  radiator  keeps  the  air  of  a  room  in  constant 
circulation;  for  the  pressure  of  the  warm  air  about  the  stove  is  always 
less  than  that  of  the  colder  and  denser  air  at  the  same  level  in  other 


CONDUCTION  AND   CONVECTION  229 

parts  of  the  room.  The  cold  air  near  the  floor  moves  slowly  toward 
this  region  of  lower  pressure,  driving  the  warm  air  upward.  As  the 
current  of  warm  air  reaches  the  ceiling,  it  spreads  out  and  crowds 
the  colder  air  downward.  The  descending  currents  are  strongest 
near  the  walls  and  windows,  especially  the  latter,  where  the  loss  of 
heat  is  most  rapid.  Cold  air,  entering  through  cracks  in  windows 
and  doorways,  provides  considerable  ventilation,  but  generally  less 
than  there  should  be.  A  fireplace  secures  excellent  ventilation  by 
maintaining  an  outward  flow  of  air  through  the  chimney. 

Currents  of  air  and  water  in  the  heating  and  ventilation  of 
buildings  are  further  considered  in  Arts.  238-241. 

193.  Winds  are  convection  currents,  due  to  the  unequal  heating 
and  cooling  of  the  atmosphere  over  different  parts  of  the  earth's 
surface.  Unequal  temperatures  at  different  places  cause  unequal 
barometric  pressures,  and  the  unequal  pressures  cause  atmospheric 
currents,  or  winds.  The  origin  of  winds  is  well  illustrated  by  the 
sea  and  land  breezes,  which  are  of  almost  daily  occurrence  along 
ocean  shores  in  temperate  and  tropical  regions.  "As  the  land  heats 
and  cools  more  quickly  than  the  sea,  it  often  becomes  warmer  than 
the  adjacent  water  during  the  day  and  cooler  at  night,  and  it  communi- 
cates its  temperature  to  the  lower  part  of  the  air.  So  by  day  the  air 
above  the  sea  is  the  denser  and  flows  toward  the  land,  and  at  night 
the  cool  air  above  the  land  flows  toward  the  sea." 

Winds,  in  general,  are  very  complex  phenomena,  and  are  influenced 
by  other  agencies  besides  temperature,  e.g.  the  percentage  of  water 
vapor  in  the  air  (which  affects  its  density),  the  direction  and  eleva- 
tion of  coast  lines  and  mountain  ranges,  and  the  rotation  of  the  earth 
on  its  axis.  As  a  rule,  the  direction  and  velocity  of  the  wind  at  any 
particular  time  and  place  are  determined  by  atmospheric  conditions 
over  an  area  many  hundred  miles  in  extent;  and  the  general  wind 
systems,  due  to  the  unequal  heating  power  of  the  sun  at  different 
latitudes,  cover  the  entire  earth.  A  full  account  of  such  matters  may 
be  found  in  any  physical  geography. 

PROBLEMS 

1.  Is  clothing  a  source  of  heat?    What  is  "warm  clothing"? 

2.  Tin  teakettles,  pots,  and  boilers  are  often  made  with  bottoms  of 
copper.     What  is  the  advantage  of  this? 


230  HEAT 

3.  The  conductivity  of  fresh  snow  is  many  times  less  than  that  of  either 
ice  or  water  (see  table).     What  is  the  reason  for  this? 

4.  Water  can  be  boiled  in  a  tray  made  of  writing  paper,  with  a  Bunsen 
flame  playing  directly  against  the  bottom  of  it.-    (Try  it.)     Why  does  the 
paper  not  burn? 

5.  Would  convection  currents  be  caused  by  cooling  a  liquid  at  the  top? 
by  cooling  it  at  the  bottom?    Would  a. piece  of  ice  cool  a  pitcher  of  warm 
water  more  or  less  quickly,  if  kept  at  the  bottom,  than  it  does  when  floating? 

6.  What  convection  currents  are  set  up  when  a  door  is  left  open  between 
a  warm  and  a  cold  room? 

7.  Does  an  open  fireplace  provide  equally  good  ventilation  whether  there 
is  a  fire  in  it  or  not? 

8.  How  does  convection  differ  from  diffusion?     Define  convection  in 
your  own  language. 

9.  On  some  days  smoke  rises  rapidly  from  chimneys,  on  others  slowly. 
Account  for  the  difference. 

10.  Criticize   the  statement  "The  air  over  a  heated  area  expands  and 
rises,  while  the  air  from  the  cooler  surrounding  regions  rushes  in  to  take  its 
place." 

11.  Inspect  at  home  the  system  of  pipes  connected  with  the  hot- water 
tank.     Note  where  the  cold  water  is  admitted  to  the  tank,  where  the  hot 
water  is  drawn  off,  and  where  the  connections  are  made  with  the  pipes  lead- 
ing to  and  from  the  heating  coil  in  the  kitchen  range.     Why  should  the  pipes 
be  placed  as  you  find  them? 

IV.  RADIATION 

194.  Radiant  Energy.  —  Near  a  large  fire  in  an  open 
grate  the  face  becomes  painfully  hot,  even  in  a  cold  room. 
If  the  hand  or  a  sheet  of  paper  is  held  before  the  face,  the 
sensation  of  intense  heat  instantly  ceases,  and  in  a  few 
seconds  the  face  becomes  cool.  Evidently  the  heat  is 
not  received  through  the  agency  of  the  air,  either  by  con- 
duction or  convection,  for  the  face  becomes  much  warmer 
than  the  air  in  contact  with  it.  Moreover,  air  is  practically 
a  non-conductor,  and  the  only  convection  current  of  heated 
air  from  the  fire  passes  up  the  chimney.  Heating  under 
such  conditions  is  the  result  of  processes  wholly  different 
from  either  conduction  or  convection. 


RADIATION  231 

These  processes  take  place  on  an  enormous  scale  in  the 
heating  of  the  earth  by  the  sun.  Let  us  recall  the  condi- 
tions under  which  this  heat  is  received.  The  average 
distance  of  the  sun  from  the  earth  is  93,000,000  miles. 
Through  all  this  distance,  to  the  earth's  atmosphere, 
there  is  neither  solid,  liquid,  nor  gas.  The  intervening 
space  is  a  perfect  vacuum,  and  where  there  are  no  mole- 
cules there  can  be  no  molecular  energy,  or  heat.  Clearly 
the  sun's  heat  does  not  make  the  long  journey  to  the  earth 
as  heat.  It  must,  however,  be  transmitted  as  some  form 
of  energy;  for  energy  in  any  form  can  become  nothing 
else  than  energy  in  some  other  form.  This  solar  energy 
passes  through  the  greater  part  of  the  earth's  atmosphere 
without  warming  it  even  to  arctic  temperatures,  but,  on 
reaching  the  earth,  it  is  transformed  into  heat  in  enormous 
quantities. 

Similar  conditions,  on  a  miniature  scale,  are  reproduced 
in  the  incandescent  electric  lamp.  When  lighted,  the  fila- 
ment is  white  hot.  The  bulb  becomes  hot  from  the  hot 
filament;  but  there  is  nothing  to  convey  heat  from  one  to 
the  other,  for  the  bulb  is  exhausted  to  a  nearly  perfect 
vacuum. 

There  is,  then,  a  form  of  energy  which  heat  becomes 
and  which  is  again  transformed  into  heat  under  certain 
conditions,  and  which  can  be  transmitted  through  a  vac- 
uum. This  is  called  radiant  energy  or  radiation.  The 
process  by  which  radiant  energy  is  transmitted  is  also  called 
radiation.  Thus  we  say  that  "radiation  takes  place  in  a 
vacuum,"  that  "a  body  loses  heat  by  radiation,"  etc. 
Radiant  energy  is  said  to  be  emitted  when  given  out  by 
bodies,  and  absorbed  when  received  by  them  and  trans- 
formed into  heat.  Other  forms  of  expression  are  com- 
mon, but  they  are  not  scientific  (Art.  201). 


2.3-2  HEAT 

195.  Light  is  Radiant  Energy.  —  At  very  high  tempera- 
tures all  bodies  give  out  light,  in  addition  to  the  radiant 
energy  which  they  emit  at  lower  temperatures.  Light  also 
travels  through  a  vacuum,  since  it  reaches  the  earth  from 
the  sun  and  the  still  more  distant  stars.  During  an  eclipse 
of  the  sun  its  heating  power  and  its  light  decrease  together 
and  are  lost  at  the  same  moment ;  as  it  reappears,  its  heat- 
ing power  is  restored.  Accurate  observation,  with  delicate 
instruments,  shows  that  the  radiant  energy  and  the  light 
of  the  sun  travel  through  space  with  the  same  velocity 
(186,000  miles  per  second).  The  reflection  of  a  beam  of 
light  by  a  plane  mirror  is  familiar.  Experiments  prove 
that  radiant  energy  is  reflected  in  the  same  manner.  Light 
and  radiant  energy  both  travel  in  straight  lines  through 
air  and  other  transparent  substances;  both  are  brought 
to  a  focus  by  a  concave  mirror  or  a  lens.  At  the  spot 
where  sunlight  is  brought  to  a  focus  by  a  large  mirror  or 
a  lens,  a  hole  is  soon  burned  through  a  piece  of  paper 
or  a  thin  board,  and  a  match  is  quickly  ignited. 

These  are  only  a  few  of  the  many  points  of  similarity 
between  light  and  radiant  energy.  In  short,  they  are  alike 
in  all  their  physical  properties;  they  are  one  and  the  same 
form  of  energy.  A  small  part  of  the  radiation  from  very 
hot  bodies  affects  the  nerve  of  the  eye,  and  produces  the 
sensation  of  sight.  This  we  call  light  or  visible  radiation. 
The  radiation  to  which  the  eye  is  not  sensitive  is  called 
invisible  or  dark  radiation. 

Light  is  "visible"  radiation  only  in  the  sense  that  it  renders 
bodies  visible.  We  do  not  see  light;  we  see  only  luminous  and  illu- 
minated bodies.  A  sunbeam  marks  its  course  across  a  darkened  room 
by  illuminating  a  cloud  of  dust  particles  in  its  track;  it  is  itself 
invisible,  and  in  dust-free  air  there  is  no  indication  whatever  of 
its  presence. 


RADIATION  233 

196.  Nature  of  Radiation.  The  Ether.  —  The  forms 
of  energy  previously  considered  do  not  exist  apart  from 
matter.  There  can  be  no  energy  of  motion  where  there 
is  nothing  to  move,  or  energy  of  strain  where  there  is  noth- 
ing to  be  compressed  or  distorted,  or  heat  where  there 
are  no  molecules.  It  is  impossible  to  conceive  of  radiant 
energy  as  an  exception  to  this  rule.  Its  existence  depends 
upon  the  existence  of  something  which  can  possess  it.  This 
something  is  called  the  luminiferous  ether,  or  simply  the 
ether.  The  ether  is  the  only  medium  or  " vehicle"  by 
which  radiant  energy  is  transmitted ;  hence  it  must  be  pres- 
ent wherever  light  travels.  It  fills  all  " empty"  space, 
from  the  intermolecular  spaces  in  ordinary  matter  to  the 
"boundless  depths  of  space"  through  which  we  receive 
light  from  the  distant  stars.  A  perfect  vacuum  is  perfectly 
full  of  ether,  and  absolutely  empty  space  is  not  found  any- 
where. 

Since  the  ether  is  not  perceived  by  any  of  the  senses,  its. 
properties  can  only  be  inferred  from  the  phenomena  to 
which  it  gives  rise.  It  is  thus  found  to  have  inertia  or  mass; 
and  hence  is  properly  called  matter.  But  it  has  not  the 
molecular  structure  of  " ordinary"  matter,  and  is  not  to 
be  regarded  as  a  highly  rarefied  gas.  The  change  of  matter 
from  one  state  to  another  (from  the  solid  to  the  liquid 
state,  from  the  liquid  to  the  gaseous,  etc.)  is  very  common; 
but,  so  far  as  we  know,  ordinary  or  molecular  matter 
never  becomes  ether,  nor  does  ether  ever  become  ordinary 
matter.  The  ether  can  neither  be  hot  nor  cold;  it  has  no 
temperature;  it  can  not  possess  heat  or  transmit  it. 

Our  first  ideas  concerning  the  nature  of  radiation  are 
derived  from  other  phenomena  which  are  in  some  respects 
similar  to  it,  and  about  which  we  have  more  direct  knowl- 
edge. A  stone,  dropped  into  a  pond,  starts  a  series  of 


234  HEAT 

waves,  which  travel  out  in  circles  from  the  center  of  dis- 
turbance. The  energy  of  the  stone  is  imparted  to  the 
water,  and  is  transmitted  by  the  waves  to  distant  parts 
of  the  pond,  perhaps  to  the  margin,  where  it  is  expended 
in  moving  grains  of  sand,  bending  blades  of  grass,  etc. 
In  such  phenomena  the  water  serves  as  a 'medium  through 
which  energy  is  transmitted  from  one  body  to  another  by 
means  of  wave  motion.  A  sounding  body,  e.g.  a  bell  or  a 
violin,  is  in  rapid  vibration,  and  sets  up  a  disturbance  in 
the  surrounding  air.  This  disturbance  travels  outward, 
or  radiates,  in  all  directions  as  a  series  of  sound  waves, 
which,  falling  upon  the  ear,  produce  the  sensation  of  sound. 
Thus  air  serves  as  a  medium  for  the  transmission  of  energy 
from  one  body  to  another  (from  the  vibrating  body  to  the 
ear),  in  the  form  of  sound  waves.  Sound  also  travels 
through  many  other  substances,  including  solids  and  liquids; 
but  it  can  not  travel  through  a  vacuum.  The  ether  is 
not  a  sound  medium. 

Similarly,  it  is  believed  that  the  vibrating  molecules  of 
ordinary  matter  set  up  a  disturbance  in  the  luminiferous 
ether,  a  disturbance  which  travels  as  a  wave  motion  with 
inconceivable  velocity  in  all  directions.  Light  consists  of 
ether  waves  which  are  of  the  right  size  or  length  to  cause 
the  sensation  of  sight,  when  they  enter  the  eye  and  fall 
upon  the  retina.  If  a  body  is  not  hot  enough  to  be  lumi- 
nous, the  ether  waves  radiated  from  it  are  too  large  or  long 
to  affect  the  retina,  and  we  call  them  "  in  visible."  Radiant 
energy,  then,  is  energy  transmitted  through  the  ether -as  a 
wave  motion.  These  waves  travel  through  the  air  and 
other  transparent  substances,  but  not  by  means  of  them. 
They  are  always  waves  in  the  ether,  which  fills  the  inter- 
molecular  spaces  in  ordinary  matter.  By  the  velocity  of 
light  we  mean  the  speed  with  which  ether  waves  travel 


RADIATION 


235 


through  space.  They  reach  the  earth  from  the  sun  in 
8  m.  and  20  sec.,  and  from  the  moon,  which  is  at  a  dis- 
tance of  240,000  mi.,  in  1.3  sec. 

197.  The  Radiometer.  —  Instruments  of  different  kinds  have  been 
invented  for  detecting  and  measuring  radiant  energy  by  the  heat 
effects  that  it  produces.     A  mercury  thermometer  with  a  coating  of 
lampblack  over  the  bulb  is  sometimes  used,  but  it  is  not  very  sensi- 
tive.    The  lampblack  absorbs  all  radiation  falling  upon  it;  and  the 
mercury  in  the  bulb  is  heated  above  the  temperature  of  the  surround- 
ing air  in  proportion  to  the  intensity  of  the  radiation. 

The  radiometer  (Fig.  183)  is  a  more  sensitive  instrument.  It 
consists  of  four  light  vanes  of  mica  or  aluminum  attached  to  a  vertical 
axis,  and  inclosed  in  a  glass  bulb  contain- 
ing air  under  very  low  pressure.  One  side 
of  each  vane  is  bright,  the  other  is  coated 
with  lampblack.  When  the  instrument  is 
placed  in  the  sunshine  or  in  the  path  of 
other  radiation,  the  vanes  rotate  with  their 
bright  side  in  advance,  the  rate  varying  with 
the  intensity  of  the  radiation.  The  rotation 
is  explained  as  follows:  The  black  surfaces 
absorb  more  radiation  than  the  bright,  and 
hence  are  warmer.  The  molecules  of  air 
that  strike  the  black  surfaces  are  heated, 
and,  rebounding  with  an  increased  velocity, 
exert  a  greater  pressure  than  the  molecules 
that  strike  the  bright  sides.  The  black  sides 
are  consequently  driven  backward.  If  the  air  in  the  bulb  were  not 
highly  rarefied,  the  collisions  among  the  molecules  would  be  so  fre- 
quent as  to  equalize  the  pressures  throughout  the  bulb,  and  the 
vanes  would  remain  at  rest. 

198.  Emission,  Absorption,  and  Reflection  of  Radiant 
Energy.  —  Bodies  emit  and  absorb  radiation  at  all  tem- 
peratures.    If  -a  body  is  warmer  than  its  surroundings,  it 
emits  more  radiation  than  it  absorbs;  if  colder,  it  absorbs 
more  than  it  emits;  if  at  the  same  temperature,  it  emits 


FIG.  183.  —  Radiometer. 


236  HEAT 

and  absorbs  radiation  in  equal  amounts.  The  hand,  held 
near  a  fire,  becomes  hot  because  it  receives  and  absorbs 
more  radiation  than  it  emits;  if  held  near  a  large  piece 
of  ice,  it  becomes  cold  because  it  receives  less  radiation 
from  the  ice  than  the  ice  does  from  it.  The  rate  at  which 
a  body  cools  by  radiation  varies  approximately  as  the 
difference  between  its  own  temperature  and  that  of  its 
surroundings.  This  is  Newton's  law  of  cooling.  For  ex- 
ample, a  cup  of  coffee  cools  five  times  as  rapidly  at  70° 
as  it  does  at  30°  in  a  room  the  temperature  of  which  is  20°. 

The  rate  of  cooling  of  a  body  by  radiation  depends  also 
upon  the  nature  of  its  surface.  Rough,  blackened  sur- 
faces are  good  radiators;  bright  and  polished  surfaces  poor 
radiators.  This  can  be  shown  by  means  of  two  vessels  of 
the  same  size  and  shape,  but  having  unlike  surfaces,  e.g. 
one  nickel-plated  and  the  other  coated  with  lampblack. 
When  filled  with  equal  quantities  of  hot  water  at  the  same 
temperature  and  allowed  to  stand  for  some  minutes,  their 
unequal  radiating  power  will  be  shown  by  the  unequal 
cooling  of  the  water  in  them. 

When  radiation  falls  upon  a  body,  part  of  it  is  absorbed, 
part  is  reflected,  and,  in  many  cases,  part  is  transmitted 
through  the  body.  Good  radiators  are  good  absorbers  of 
radiation;  and  poor  radiators,  poor  absorbers.  Lampblack 
is  the  best  radiator  and  the  best  absorber  known.  It 
absorbs  practically  all  radiation,  both  visible  and  invisible, 
that  falls  upon  it.  Any  polished  metal  reflects  much  the 
greater  part  of  all  radiation,  and  absorbs  the  remainder. 
The  absorption  of  invisible  radiation  is  shown  by  the  heat- 
ing effects  produced;  the  absorption  of  light  by  the  color 
effects.  A  white  surface  reflects  nearly  all  light  that  falls 
upon  it;  a  black  surface  reflects  almost  none;  a  colored  sur- 
face reflects  part  and  absorbs  part.  Light  becomes  heat 


RADIATION  237 

when  absorbed;  but  its  energy  is,  in  general,  very  small 
compared  with  that  of  the  longer  waves  of  dark  radiation. 
Since  only  the  unabsorbed  radiation  can  be  reflected, 
it  is  obvious  that  good  absorbers  are  poor  reflectors,  and 
•vice  versa.  Reflected  radiation  does  not  affect  the  tem- 
perature of  the  body  upon  which  it  falls. 

199.  Selective    Transmission    and    Absorption.  —  Sub- 
stances   that    transmit    light    are    called    transparent    or 
translucent;  those  that  do  not  are  called  opaque.     Some 
transparent  substances  also  transmit  the  longer  waves  of 
dark  radiation;  others  do  not.   Clear  glass  transmits  all  light 
waves  and  a  considerable  part  of  the  radiation  from  bodies 
nearly  red-hot,  but  almost  completely  absorbs  the  longer 
waves  which  radiate  from  bodies  at  ordinary  temperatures. 
A  large  part  of  the  sun's  radiation  passes  through  window- 
glass,  and,  in  a  sunny  room,  is  absorbed  by  the  objects 
upon  which  it  falls.     But,   as  the  radiation  from  these 
bodies  can  not  penetrate  glass,  the  energy  is  trapped  in  the 
room,  which  may  thus  become  much  warmer  than  the  air 
outside.     Water  is  another  highly  transparent  substance 
which  transmits  very  little  dark  radiation.     On  the  other 
hand,  substances  which  transmit  dark  radiation  are,  in 
some  cases,  perfectly  opaque  to  light.     A  solution  of  iodine 
in  carbon  disulphide  is  an   example   (Laboratory  Exercise 
26).     Rock  salt  and  pure,  dry  air  are  good  transmitters  of 
both  light  and  dark  radiation. 

Substances  that  transmit  dark  radiation  are  sometimes 
called  diathermanous  and  those  that  do  not,  athermanous. 
The  unequal  absorption  of  ether  waves  of  different  lengths 
by  the  same  substance  is  called  selective  absorption. 

200.  The  Heating  of  the  Atmosphere.  —  The  pure,  dry  air  at 
high  altitudes  transmits  solar  radiation  in  enormous  quantities  and 
absorbs  very  little.     Aeronauts  find  the  atmosphere  intensely  cold 


238  HEAT 

above  a  height  of  three  or  four  miles.  As  the  radiation  approaches 
the  earth,  the  rate  of  absorption  rapidly  increases,  principally  on 
account  of  the  greater  amount  of  water  vapor;  for  experiments  have 
shown  that  the  absorbing  power  of  air  containing  the  average  amount 
of  water  vapor  is  seventy- two  times  as  great  as  that  of  perfectly  dry 
air  of  the  same  density.  The  absorption  at  lower  levels  is  further 
increased  by  the  dust  particles  in  the  air. 

Elaborate  investigations  conducted  at  the  Smithsonian  Institu- 
tion, Washington,  D.C.,  and  at  the  solar  observatory  on  Mt.  Wilson, 
in  southern  California,  have  shown  that,  on  the  average,  the  atmos- 
phere transmits  about  two  thirds  of  the  solar  energy.  Much  of  this 
is  absorbed  by  the  surface  of  the  land  and  the  ocean;  the  remainder 
is  reflected.  The  reflected  radiation  is  partly  absorbed  on  its  way 
out  through  the  atmosphere.  The  absorbed  radiation  warms  the 
surface  of  the  land,  and  this  in  turn  warms  the  air  in  contact  with  it. 
This  is  especially  noticeable  on  a  hot  day  in  summer,  when,  if  there 
is  no  wind,  the  air  close  to  the  ground  is  many  degrees  warmer  than 
at  a  height  of  a  few  feet.  The  heating  of  the  air  at  the  bottom  causes 
convection  currents  (winds),  by  which  the  heat  is  carried  to  consider- 
able altitudes;  but  the  temperature  is  necessarily  the  highest  at 
the  source  of  the  heat,  i.e.  at  the  earth's  surface. 

The  earth  is  cooled  at  night  principally  by  radiation.  The  loss 
of  heat  is  rapid  on  clear  nights,  especially  when  the  atmosphere  is 
very  dry.  A  moist  or  cloudy  atmosphere  absorbs  the  radiation, 
thus  serving  as  a  blanket  to  the  earth.  Hence  clear  nights  are,  as 
a  rule,  the  coldest.  At  high  altitudes,  where  there  is  but  little  hin- 
drance to  radiation  either  by  day  or  night,  sheltered  valleys  are  quickly 
warmed  in  summer  by  the  early  morning  sunshine;  and  a  sudden  chill 
follows  the  disappearance  of  the  sun  in  the  evening,  the  nights  being 
often  cold  enough  for  frost. 

Thus  we  see  that  the  atmosphere,  or  rather  the  moisture  in  it, 
renders  a  very  important  service  in  moderating  the  intensity  of  solar 
radiation  by  day  and  in  retaining  the  earth's  heat  by  night. 

201.  Summary  on  the  Transmission  of  Heat.  —  Heat  is 
transmitted  from  one  portion  of  matter  to  another  by  con- 
duction only;  it  is  transmitted  from  one  place  to  another 
through  matter  by  conduction,  and  with  matter  by  convec- 


RADIATION  239 

tion.  A  body  loses  heat  by  conduction  and  by  the  emis- 
sion of  radiant  energy;  it  gains  heat  by  conduction  and  by 
the  absorption  of  radiant  energy. 

Before  the  true  nature  of  radiant  energy  was  known, 
dark  radiation  was  supposed  to  be  a  form  of  heat,  and 
was  called  "  radiant  heat."  According  to  this  older  view, 
bodies  " radiate  heat"  and  " absorb  heat"  and  "heat  is 
transmitted  by  radiation."  This  is  still  the  popular  lan- 
guage of  the  subject,  but  it  is  disappearing  from  scientific 
literature. 

PROBLEMS 

1.  Why  does  snow  melt  more  quickly  when  covered  with  a  thin  layer 
of  earth? 

2.  Why  is  light-colored  clothing  more  comfortable  in  summer  than 
black? 

3.  Why  is  the  difference  between  the  temperature  in  the  sunlight  and  in 
the  shade  greater  upon  the  top  of  a  mountain  than  at  a  low  elevation? 

4.  Why  must  those  who  climb  snow-covered  mountains  take  special  care 
to  protect  their  faces? 

5.  Why  is  the  heating  power  of  the  sun  less  at  morning  and  evening 
than  it  is  during  the  middle  of  the  day?    Why  less  in  winter  than  in  summer? 

6.  What  is  the  "solar  water-heater"?     Explain  its  action. 

7.  The  moon  has  no  atmosphere  and  its  days  and  nights  are  each  two 
weeks  long.     What  effect  do  you  think  these  conditions 

must  have  on  the  temperature  of  lunar  days  and  nights? 

8.  A  Dewar  flask  (Fig.  184)  for  containing  liquid  air 
consists  of  "a  double- walled  glass  vessel,  the  space  be- 
tween the  walls  being  exhausted  as  completely  as  possible. 
Traces  of  mercury  vapor  are  left  in  this  space;  and  at 
a  low  temperature  this  freezes,  forming  a  metallic  surface 

over  the  glass  walls."     The  access  of  heat  to 

liquid  air  in  such  a  flask  is  almost  completely 

3       Dewar  Flask, 
prevented.     Explain. 

NOTE. — The  commercial  form  of  the  Dewar  flask  is  known  as 
the  "thermos  bottle"    (Fig.    i84a).     The  double-walled  glass 
bottle  is  inclosed  in  a  nickel-plated  metal  case.     What  useful 
FIG  Ts  £_PurP°se  is  served  by  the  nickel  plating?     It  is  claimed  that  the 
Thermos    thermos  bottle  keeps  liquids  hot  24  hours  in  the  coldest  weather, 
Bottle,     and  ice-cold  liquids  ice  cold  72  hours  in  the  hottest  weather. 


240  HEAT 

V.   CHANGES  IN  VOLUME  AND  PRESSURE 

202.  Linear  Expansion  of  Solids.  —  With  few  excep- 
tions, none  of  which  are  important,  solids  expand  when 
heated  and  contract  when  cooled.  The  rate  of  expansion 
is  different  for  bodies  of  different  material;  but  it  is  in  all 
cases  so  small  that  some  special  device  must  be  employed 
in  order  to  measure  with  any  degree  of  accuracy  the  in- 
crease in  the  length  of  even  a  long  rod  when  its  tempera- 
ture is  raised  many  degrees.  The  methods  by  which  this 
is  done  are  best  studied  in  the  laboratory.  (See  Labora- 
tory Exercise  27.) 

A  solid  expands  in  each  of  its  three  dimensions,  and 
increase  in  any  one  of  them  is  called  linear  expansion.  In 
most  cases  it  is  only  change  of  length  that  is  important. 
The  increase  in  length  per  unit  length,  for  a  rise  of  tempera- 
ture of  one  degree,  is  called  the  coefficient  of  linear  expan- 
sion of  the  substance.  This  coefficient  may  equally  well 
be  regarded  as  the  ratio  of  the  whole  increase  in  length  to 
the  whole  length  of  the  body,  for  a  rise  of  temperature  of 
one  degree. 

For  example,  the  coefficient  of  linear  expansion  of  steel  is  .000012. 
This  means  that  each  centimeter  of  length  of  a  steel  bar  or  rod  in- 
creases to  i. 000012  cm.  or  each  foot  of  its  length  to  1.000012  ft., 
with  a  rise  of  temperature  of  one  degree  Centigrade.  The  coefficient 
for  aluminum  is  .000023,  or  approximately  twice  that  of  steel. 

The  expansion  of  a  solid  with  any  rise  of  temperature 
can  readily  be  computed  when  its  length  and  its  coefficient 
of  expansion  are  known.  This  is  a  matter  of  great  impor- 
tance to  engineers  and  builders. 

EXAMPLE.  What  is  the  expansion  of  a  thirty-foot  steel  rail  between 
a  minimum  winter  temperature  of  —  30°  C.  and  a  maximum  summer 
temperature  of  40°?  The  expansion  per  foot  per  degree  is  .000012  ft. 
Hence  the  total  expansion  is  30  X  70  X  .000012  =  .025  ft.  or,3  in. 


CHANGES    IN    VOLUME    AND    PRESSURE         241 

COEFFICIENTS  OF  LINEAR  EXPANSION 

Hard  rubber 000084        Platinum   0000088 

Zinc 0000294       Glass 0000086 

Lead 0000286       Oak  parallel  to  grain 000005 

Aluminum   000023  Steel     alloyed    with 

Brass QOQOiSS  36%  nickel 0000009 

Copper 0000172       Porcelain  (Berlin) 0000027 

Iron  and  steel 000012         Quartz,  fused 0000004 

203.   Effects    and    Applications    of    Expansion.  —  The 

expansion  and  contraction  of  solids  with  a  change  of  tem- 
perature enters  into  the  .affairs  of  daily  life  in  many  ways. 
Frequently  it  presents  itself  merely  as  a  troublesome  fact 
which  must  be  taken  into  account.  Thus  it  is  often  neces- 
sary to  make  special  provision  for  expansion  in  designing 
metal  structures  and  machinery.  A  steel  truss  bridge 
would  wreck  its  foundations  with  its  change  of  length  be- 
tween winter  and  summer,  if  it  were  rigidly  anchored  to 
them ;  hence  one  end  is  supported  upon  steel  rollers.  Bridge 
engineers  allow  for  an  expansion  of  one  inch  in  each  80  ft. 
of  length,  in  localities  where  the  changes  of  temperature 
are  extreme.  The  rails  of  tracks  are  laid  with  a  small 
space  between  their  ends,  which  provides  room  for  expan- 
sion. The  change  of  length  of  steam  and  hot- water  pipes 
is  provided  for  by  inserting  expansion  joints,  at  which  the 
end  of  one  pipe  is  free  to  slip  back  and  forth  within  the  end 
of  a  larger  one.  Bends  in  the  pipes  can  often  be  made 
to  serve  for  the  same  purpose;  for  a  bend  will  yield  more 
or  less  as  the  length  of  the  pipe  changes. 

Referring  to  the  table  of  expansion  coefficients,  it  will 
be  seen  that  the  expansion  of  steel  containing  36%  of  nickel 
is  much  the  smallest  in  the  list,  being  less  than  one  fifth 
that  of  oak  and  about  one  tenth  that  of  glass.  This  is  a 
very  valuable  property.  "  Invar  steel  is  a  nickel  steel  in 
which  by  mechanical  treatment  the  coefficient  has  been  still 


242  HEAT 

further  reduced.  It  is  used  extensively  for  the  construc- 
tion of  standards  of  length,  steel  tapes,  pendulums,  etc." 
Expansion  is  usefully  applied  in  various  ways.  The 
wooden  wheels  of  vehicles  have  iron  tires,  which  are  made 
just  large  enough  to  slip  on  when  heated;  and,  in  cooling, 
they  contract  so  as  to  fit  tightly.  Large  cannon  are  some- 
times reenforced  with  an  outer  casing  or  jacket  of  steel, 
which  is  shrunk  on  like  the  tire  of  a  wheel.  Red-hot  rivets 
are  used  in  joining  the  steel  plates  of  tanks  and  boilers; 
and,  by  their  contraction  in  cooling,  they  draw  the  plates 
together  with  great  force. 

Various  mechanical  devices  have  been  invented,  whose  action 
depends  upon  the  unequal  expansion  of  different  metals.  The  princi- 
ple involved  can  be  illustrated 
with  a  compound  bar  consisting 
of  a  strip  of  brass  and  one  of 
iron,  riveted  together  in  several 
places  (Fig.  185).  The  bar  is 
straight  or  nearly  so  at  ordi- 
nary temperatures;  but,  when 
heated,  it  takes  the  shape  of  a  circular  arc,  with  the  brass  strip 
on  the  outside  of  the  curve.  The  bending  is  due  to  the  greater 
expansion  of  the  brass  (see  table  of  coefficients).  The  bar  becomes 
straight  again  on  cooling,  for  the  brass  then  contracts  more  than 
the  iron.  Metallic  thermometers  are  constructed  on  this  principle. 
Sometimes  they  are  self-recording,  as  shown  in  Fig.  186.  A  compound 
strip  of  brass  and  steel  is  formed 
into  a  spiral,  with  the  brass  on  the 
outside.  When  the  temperature 
rises,  the  spiral  becomes  more 
curved  and  its  outer  end  moves 
upward,  raising  the  end  C  of  a 
lever;  when  the  temperature  falls, 
the  lever  is  depressed.  A  pen  at 


FIG.  185.  —  Compound  Bar  Showing 
Unequal  Expansion. 


FIG.  186.  —  Metallic  Thermometer, 
Recording. 


the  end  of  the  lever  records  its  movements  on  a  paper  wrapped 
round  the  drum  D,  which  is  moved  by  clockwork. 


CHANGES  IN  VOLUME  AND  PRESSURE          243 

204.  The  Expansion  of  Liquids.  —  In  the  expansion  of 
liquids  and  gases  it  is  increase  of  volume,  or  cubical  expan- 
sion, with  which  we  are  concerned.  The  coefficient  of 
cubical  expansion  of  a  liquid  (or  a  solid)  is  the  increase  in 
its  volume  per  unit  volume,  for  a  rise  of  temperature  of 
one  degree. 

As  usually  contrived  (Fig.  170),  experiments  on  the 
expansion  of  liquids  give  their  apparent  expansion,  i.e.  the 
difference  between  their  true  expansion  and  the  expansion 
of  the  containing  vessel.  The  true  expansion  of  a  liquid 
is  the  sum  of  its  apparent  expansion  and  the  cubical  expan- 
sion of  the  material  of  the  containing  vessel.  Liquids 
differ  from  one  another  in  their  rates  of  expansion;  and,  in 
general,  they  expand  more  rapidly  at  higher  temperatures. 
Mercury  is  exceptional  in  that  its  expansion  is  practically 
uniform  between  o°  and  100°,  hence  the  degree  intervals 
on  a  mercury  thermometer  are  equally  spaced. 

The  coefficient  of  cubical  expansion  of  a  solid  is  (very 
approximately)  three  times  its  coefficient  of  linear  expan- 
sion, since  the  rate  of  expansion  is  the  same  in  the  three 
dimensions.  Multiplying  the  linear  coefficients  in  the  table 
for  solids  by  three,  a  comparison  can  be  made  with  the 
coefficients  for  liquids  given  in  the  table  below.  It  will 
be  found  that  the  expansion  of  liquids  is,  in  general,  much 
the  greater.  The  total  expansion  of  water  between  4° 
and  100°  is  a  little  over  4%. 

COEFFICIENTS  OF  CUBICAL  EXPANSION 

Ether    0.0018  Turpentine 0.0007 

Alcohol  (5°  to  6°) 0.00105  Glycerin 0.0005 

Alcohol  (49°  to  50°) 0.00122  Water  (5°  to  6°) 0.000022 

Acetic  acid   0.00105  Water  (49°  to  50°) 0.00046 

Petroleum 0.0009  Water  (99°  to  100°) 0.00076 

Olive  oil 0.0008  Mercury 0.00018 


244  HEAT 

205.  Expansion  of  Water.  —  The  expansion  of  water  is  curiously 
irregular.     It  contracts  as  its  temperature  rises  from  o°  to  4°.     When 
heated  beyond  this  point  it  begins  to  expand,  at  first  very  slowly, 
then  more  and  more  rapidly  (see  table).     Hence  the  density  of  water 
is  greatest  at  4°  C.  (about  39°  R). 

This  behavior  of  water  is  of  great  importance  in  the  economy  of 
nature.  In  winter  the  water  of  a  lake  loses  heat  at  the  surface  by 
contact  with  the  cold  air  and  by  radiation.  As  the  water  at  the 
surface  cools,  it  becomes  denser  and  sinks,  displacing  the  water  at 
the  bottom.  This  continues  until  the  water  is  cooled  throughout 
to  a  temperature  of  4°.  With  further  cooling  of  the  surface  layer, 
it  expands  and  remains  at  the  top.  Hence  freezing  takes  place  at 
the  surface,  while  the  water  at  only  a  slight  depth  is  at  4°.  Further 
loss  of  heat  takes  place  only  by  conduction,  which  is  slow  in  both 
water  and  ice;  hence  lakes  and  streams  freeze  only  to  a  depth  of  a 
few  feet,  even  in  a  long,  cold  winter,  and  the  fish  and  other  inhabit- 
ants of  the  waters  are  not  destroyed. 

206.  Expansion  of  Gases.     Laws  of  Gay-Lussac    and 
Charles.  —  We  have  seen  that  different  solids  and  differ- 
ent liquids  expand  at  very  unequal  rates  when   heated. 
The  rate  of  expansion  of  gases,  on  the  contrary,  is  found 
to  be  the  same  for  all,  to  a  very  close  approximation,  at 
all  temperatures  and  at  any  constant  pressure,  provided 
only  that  the  temperature  is  considerably  above  that  at 
which  the  gas  liquefies  under  the  given  pressure.     The 
volume  of  a  given  mass  of  any  gas  increases,  under  constant 
pressure,  by  ^js  or  ^s  volume  at  o°  C.  for  each  degree  of  rise 
in  temperature.     This  is  known  as  the  law  of  Gay-Lussac  or 
•the  law  of  Charles,  after  two  French  physicists  who  shared 
in  its  discovery.     The  fraction  ^js  or  -003665  is,  according 
to  the  law,  the  coefficient  of  cubical  expansion  of  all  gases. 
This  coefficient  is  much  larger  than  that  of  liquids  and 
solids  in  general.     (See  table.) 

When  a  gas  is  heated  without  being  permitted  to  expand, 
its  pressure  increases  by  ^73-  of  the  pressure  at  o°  C.  for 
each  degree  of  rise  in  temperature.  This  is  more  properly 


CHANGES  IN  VOLUME  AND   PRESSURE 


245 


called  the  law  of  Charles.  It  is  a  necessary  consequence 
of  the  law  of  Gay-Lussac  and  the  law  of  Boyle  (Art.  54), 
taken  together.  For,  if  a  gas  is  heated  at  constant  pres- 
sure, it  expands  according  to  the  law  of  Gay-Lussac.  If 
it  is  then  compressed  to  its  original  volume  without  change 
of  temperature,  the  pressure  increases  according  to  Boyle's 
law.  It  is  then  in  the  condition  in  which  it  would  be  if  it 
had  been  heated  under  constant  volume;  i.e.  the  net  result  is 
an  increase  of  pressure,  in  agreement  with  the  law  of  Charles. 

The  following  are  numerical  examples  illustrating  the  laws:  If 
a  mass  of  gas  is  heated  under  constant  pressure  from  o°  to  10°,  its 
increase  of  volume  is  ^W  of  its  original  volume,  and  its  volume  is 
then  1M  of  its  volume  at  o°.  If  it  is  cooled  from  o°  to  —  50°,  its 
loss  of  volume  is  3%  of  its  volume  at  o°,  and  its 
volume  is  then  ff  |  of  its  volume  at  o°.  If  it  is 
heated  at  constant  volume  from  o°  to  100°,  its  pres- 
sure becomes  f  y§  as  great  as  at  o°. 

207.  Absolute  Temperature  and  Absolute  Zero.  — 
Let  VQ  denote  the  volume  of  a  body  of  gas  at  o°  C., 
and  vi  its  volume  at  any  other  temperature  t\  under 
the  same  pressure.  The  increase  in  its  volume  is 

+5    then  v\  —  VQ,  and  this  increase  is  --  of  the  volume 

273 

at  o°;   that  is, 


125 

- 

400 

100 

.- 

375 
373 

75 

350 

50 

325 

25 

.- 

300 

0 

.- 

275 
273 

3-25 

250  ^ 

w-50 

225  | 

o 

w 

|!  -75 

200  ^ 

3-100 

-125 

_ 

175  | 
150 

-150 

126 

-175 

.- 

100 

-200 

.  - 

75 

-225 

.- 

50 

-250 

.  - 

25 

-273 
FIG 

.  i 

0 

57- 

) 


From  which 


A 
273 


(i) 


Similarly,  if  v<z  denotes  the  volume  of  the  gas  at 
temperature  h,  under  the  same  pressure,  then  — 


=  V0 


273 


(2) 


246  HEAT 

Dividing  the  members  of  equation  (i)  by  the  corresponding  mem- 
bers of  equation  (2),  we  have 

vi  "273 


*>i      273  +  h 
which  reduces  to  -  =  ^  +  fa-  (3) 

The  relation  expressed  by  equation  (3)  has  led  to  the  adoption  of  a 
temperature  scale  whose  degrees  are  the  same  as  those  of  the  Centi- 
grade scale  but  whose  zero  is  at  —  273°  C.  This  scale  of  temperature 
is  called  the  absolute  scale,  and  its  zero  the  absolute  zero.  The 
freezing  point  is  273°  Abs.  and  the  boiling  point  373°  Abs.  Any 
temperature  on  the  Centigrade  scale  is  changed  to  the  absolute  scale 
by  adding  273.  If  we  let  T  denote  temperatures  on  the  absolute 
scale,  then  TI  =  273  +  /i,  and  T%  =  273  +  /2,  and  equation  (3) 
becomes  — 


That  is:  Under  constant  pressure  the  volume  of  any  body  of  gas  is  pro- 
portional to  its  absolute  temperature.  This  is  the  law  of  Gay-Lussac, 
stated  in  its  simplest  form. 

If  the  law  held  for  all  temperatures,  it  is  evident  that  at  absolute 
zero  the  volume  of  any  mass  of  gas  would  be  zero;  but  no  substance 
exists  as  a  gas  at  absolute  zero.  All  known  gases  have  been  liquefied 
and  all  except  helium  reduced  to  the  solid  state  at  temperatures  above 
absolute  zero;  and,  as  stated  before,  the  law  does  not  express  the 
behavior  of  gases  when  near  the  point  of  condensation. 

If  in  the  above  equations  we  substitute  pressures  for  volumes, 
we  have  the  relations  that  hold  between  the  pressure  and  the  tem- 
perature of  a  gas,  at  constant  volume;  and  we  arrive  at  the  conclusion 
that  the  pressure  of  a  gas  is  proportional  to  its  absolute  temperature, 
the  volume  remaining  constant.  This  law  holds  for  any  gas  as  it  is 
cooled,  until,  at  a  certain  low  temperature,  the  molecules  begin  to 
cohere,  when  the  loss  of  pressure  becomes  more  rapid  than  the  law 
indicates.  If  it  were  not  for  cohesion,  we  should  expect  the  law  to 


CHANGES  IN  VOLUME  AND  PRESSURE          247 

hold  at  all  temperatures,  in  which  case  the  pressure  would  vanish 
only  at  absolute  zero.  Since  pressure  is  due  to  molecular  motion, 
the  molecules  would  then  be  at  rest,  and  the  gas  would  have  lost  all 
its  heat  energy.  By  other  lines  of  reasoning  which  belong  to  more 
advanced  physics,  it  is  proved  that  the  absolute  zero  is  indeed  what 
its  name  indicates,  namely,  the  temperature  at  which  a  body  would 
possess  no  molecular  kinetic  energy,  or  no  heat.  No  substance  has 
yet  been  cooled  to  absolute  zero;  but  this  temperature  has  been  more 
and  more  closely  approached  in  recent  years.  By  rapid  evaporation 
in  a  vacuum,  solid  hydrogen  has  been  cooled  to  -260°  C.  or  13°  Abs., 
and  in  1908  helium  was  liquefied  at  a  temperature  estimated  at  5° Abs. 


PROBLEMS 

1.  The  thinner  a  glass  tumbler  is,  the  less  likely  it  is  to  break  when  hot 
water  is  poured  into  it.    Why? 

2.  Why  can  not  an  air  thermometer  be  used  for  measuring  the  lowest 
attainable  temperatures? 

3.  In  all  accurate  work  the  reading  of  a  barometer  must  be  "corrected 
for  temperature";    i.e.  its  true  height  is  taken  as  the  height  at  which  it 
would  stand  if  the  temperature  of  the  mercury  were  o°  C.     A  barometer 
reading  is  75.6  cm.  at  a  temperature  of  22°;  find  its  true  or  corrected  height. 

4.  The  steel  cables  of  the  Manhattan  suspension  bridge  in  New  York 
City  are  about  1475  ft-  long  between  the  towers.     How  much  does  their 
length  change  in  this  span  between  winter  and  summer,  allowing  a  minimum 
winter  temperature  of  -25°  C.  and  a  maximum  summer  temperature  of  35°? 

6.  To  what  temperature  must  a  gas  be  heated,  under  constant  pressure, 
in  order  to  double  its  volume,  the  temperature  at  the  start  being  30°  C.? 

6.  A  body  of  gas  at  10°  and  a  pressure  of  one  atmosphere  is  inclosed  in  a 
vessel  and  heated  to  300°,  none  of  the  gas  being  allowed  to  escape.     What 
is  the  pressure  at  that  temperature? 

7.  If  a  balloon  at  the  beginning  of  its  ascent  is  fully  inflated  with  gas  at 
20°,  what  fractional  part  of  the  gas  must  be  allowed  to  escape  in  rising  to  a 
height  where  the  pressure  is  reduced  one  half  and  the  temperature  is  -io°? 

Suggestion.  —  The  pressure  inside  the  balloon  is  practically  the  same  as 
that  of  the  surrounding  air.  Pressure  and  temperature  really  change 
together;  but  the  final  result  is  the  same  as  if  they  took  place  separately. 
Hence  compute  first  the  effect  of  change  of  temperature,  assuming  constant 
pressure,  then  the  effect  of  change  of  pressure. 


248  HEAT 

VI.  MEASUREMENT  OF  HEAT.     SPECIFIC  HEAT 

208.  The  Heat  Unit.  —  Heat  being  a  form  of  energy,  it 
can  be  measured  in  terms  of  any  of  the  units  by  which 
mechanical  energy  is  measured  (foot-pound,  etc.) ;  they  are 
not  used,  however,  as  there  are  more  convenient  units 
for  the  purpose.     Two  heat  units  are  in  common  use:  one, 
the  calorie,  is  the  amount  of  heat  required  to  raise  the  tem- 
perature of  one  gram  of  water  one  degree  Centigrade ;  the 
other  is  the  amount  of  heat  required  to  raise  the  tempera- 
ture of  one  pound  of  water  one  degree  Fahrenheit.     The 
calorie  is  almost  exclusively  used  in  scientific  work,  and 
is  the  only  heat  unit  used  in  this  book. 

The  heat  received  or  given  out  by  any  mass  of  water, 
when  it  is  warmed  or  cooled  through  any  range  of  tempera- 
ture, is  measured  by  the  product  of  its  mass  and  its  change 
of  temperature. 

For  example,  to  warm  10  g.  of  water  one  degree  requires  10  calories; 
to  warm  10  g.  from  8°  to  63°  requires  10  X  (63  —  8)  =  550  calories. 
In  cooling  from  63°  to  8°,  10  g.  of  water  would  give  out  550 
calories. 

The  amount  of  heat  required  to  raise  the  temperature  of  one  gram 
of  water  one  degree  is  not  exactly  the  same  at  all  temperatures,  but 
the  difference  is  too  small  to  be  of  importance  except  in  the  most 
accurate  work.  The  numerical  relation  between  the  calorie  and  the 
units  of  mechanical  energy  is  considered  in  Art.  242. 

209.  Specific  Heat.  —  When  equal  masses  of  hot  and 
cold  water  are  mixed,  the  resulting  temperature  is  midway 
between  the  original  (initial)  temperatures  of  the  separate 
masses;  e.g.  if  the  temperature  of  the  cold  water  is  20° 
and  that  of  the  hot  water  100°,  the  temperature  of  the  mix- 
ture will  be  60°.     In  cooling  one  degree,  the  hot  water 
gives  out  enough  heat  to  warm  the  equal  mass  of  cold  water 
one  degree. 


MEASUREMENT  OF  HEAT.     SPECIFIC  HEAT      249 

If  a  piece  of  hot  metal  is  dropped  into  an  equal  mass  of 
cold  water,  the  resulting  temperature  is  far  below  the 
average  of  the  initial  temperatures.  If  the  metal  is  iron, 
for  example,  it  will  be  found  that  the  temperature  of  the 
iron  falls  about  9  degrees  for  each  degree  of  rise  in  the  tem- 
perature of  the  water.  Plainly,  therefore,  iron  gives  out 
only  ^  as  much  heat  as  an  equal  mass  of  water  does  during 
an  equal  fall  of  temperature.  It  follows  further  that  only  ^ 
as  much  heat  is  required  to  raise  the  temperature  of  a  mass 
of  iron  a  given  number  of  degrees  as  is  required  to  raise 
the  temperature  of  an  equal  mass  of  water  the  same  num- 
ber of  degrees.  When  copper  or  zinc  is  used  in  the  experi- 
ment, the  ratio  is  found  to  be  approximately  ^T;  with  lead 
it  is  about  3*0- 

The  ratio  of  the  quantity  of  heat  required  to  warm  any 
mass  of  a  substance  one  degree  to  the  quantity  required 
to  warm  an  equal  mass  of  water  one  degree  is  called  the 
specific  heat  of  the  substance.  (Compare  with  the  defini- 
tion of  specific  gravity.)  The  specific  heat  of  a  substance 
is  numerically  equal  to  the  number  of  calories  required  to 
raise  the  temperature  of  one  gram  of  the  substance  one 
degree  Centigrade.  (Why?) 

The  number  of  calories  required  to  warm  any  mass  of 
a  substance  through  any  number  of  degrees  is  measured 
by  the  product  of  the  mass  of  the  body,  its  rise  of  tempera- 
ture, and  its  specific  heat.  (Why?) 

The  specific  heat  of  water  is  unity,  by  definition;  it  is 
very  large  compared  with  that  of  most  other  substances, 
especially  the  metals,  and  is  exceeded  only  by  hydrogen. 
In  the  following  table  the  substances  are  named  in  the  order 
of  their  specific  heats.  Note  that  the  specific  heat  of 
water  changes  with  a  change  of  state,  its  value  for  ice 
being  .504  and  for  steam  .48. 


250  HEAT 

TABLE  OF  SPECIFIC  HEATS 

Hydrogen  (at  constant  pressure)  3.409         Aluminum 0.218 

Water    i.ooo          Glass    0.198 

Alcohol  (o°  to  40°)     0.597          Iron    0.113 

Ice   0.504          Brass  and  copper 0.094 

Steam    0.480         Zinc    0.094 

Air  (at  constant  pressure)    0.237          Mercury 0.033 

Marble 0.216         Lead 0.031 

210.  Measurement  of  Specific  Heat.  —  The  method 
generally  employed  for  determining  the  specific  heat  of  a 
substance  is  known  as  the  "method  of  mixtures."  It  is 
illustrated  by  the  following  example:  A  brass  calorimeter 
weighing  100  g.  contains  400  g.  of  water  at  18°.  Into  this 
is  put  a  roll  of  sheet  iron,  weighing  190  g.  and  heated 
to  1 00°.  After  stirring,  the  temperature  of  the  water  is 
22°,  and  this  is  assumed  to  be  the  temperature  of  the  roll 
of  iron  and  the  calorimeter.  The  specific  heat  of  the 
calorimeter  is  given  as  .094.  The  specific  heat  of  iron  is 
to  be  found  from  the  experimental  data,  and  is  denoted 
by  s.  The  computation  is  as  follows: 

Rise  of  temp,  of  calorimeter  and  water    =  22°  —  18°  =          4° 
Heat  received  by  the  calorimeter   =  100  X  4  X  .094  =        37.6  cal. 
Heat  received  by  the  water  =  400  X  4  =      1600    cal. 

Fall  of  temperature  of  the  iron  =  100°  —  22°  =        78° 

Heat  given  out  by  the  iron  =  190  X  78  X  s  =.  14,820  s  cal. 

Assuming  that  the  transfers  of  heat  take  place  only 
among  the  calorimeter  and  its  contents,  it  follows  that 
the  heat  given  out  by  the  iron  in  cooling  to  the  tempera- 
ture of  the  " mixture"  is  equal  to  the  heat  gained  by  the 
calorimeter  and  water  in  coming  to  the  same  temperature; 
that  is, 

14,8205  =  37.6  +  1600; 

from  which  s  =  1637.6  -f-  14,820  =  .no. 


MEASUREMENT  OF  HEAT.     SPECIFIC  HEAT     251 

211.  The  Heat  Equation.  —  The  above  example  illus- 
trates the  method  of  treating  the  experimental  data  in  all 
experiments    in  calorimetry   (the    measurement  of   heat) 
and  in  the  solution  of  problems.     The  following  summary 
of  the  method  will  therefore  be  of  service  now  and  later. 

1.  Find  numerical  or  algebraic  expressions  for  the  sep- 
arate quantities  of  heat  received  or  given  out  by  the  differ- 
ent bodies  (including  the  vessel)  during  the  equalization 
of  temperature. 

2.  Write  the  sum  of  the  quantities  of  heat  given  out 
equal  to  the  sum  of  the  quantities  of  heat  received.     This 
is  the  heat  equation. 

3.  The  heat  equation  contains  as  an  unknown  quantity 
the  quantity  sought  (specific  heat,  heat  of  fusion,  or  heat 
of  vaporization).     To  find  this  quantity,  solve  the  equa- 
tion by  the  usual  algebraic  processes. 

212.  The  Control  of  Heat  in  Calorimetric  Experiments.  —  Any 
transfer  of  heat  between  the  contents  of  the  calorimeter  and  the  sur- 
rounding air  or  other  bodies  during  an  experiment  is  a  source  of  error, 
and  is  to  be  avoided  as  far  as  possible.     The  calorimeter  is    sually 
nickel-plated  and  brightly  polished  to  diminish  radiation  when  it  is 
warmer  than  the  surrounding  air,  and  to  diminish  absorption  when 
it  is  cooler.     It  should  stand  on  a  poor  conductor  (wood)  and  should 
be  touched  with  the  hands  as  little  as  possible,  to  avoid  conduction 
to  or  from  the  hand.     A  calorimeter  is  like  a  leaky  vessel.     By  such 
precautions  as  these  we  endeavor  to  stop  up  the  leaks. 

At  the  beginning  of  an  experiment  the  water  should  be  taken  at 
such  a  temperature  that  it  (and  the  calorimeter)  will  be  colder  than 
the  air  during  a  part  of  the  time  and  warmer  during  a  part,  in  order 
that  the  gain  of  heat  by  conduction  and  absorption  at  the  lower 
temperature  may  be  as  nearly  as  possible  equal  to  the  loss  by  conduc- 
tion and  radiation  at  the  higher  temperature. 

For  accurate  work  other  and  much  more  elaborate  precautions 
than  these  are  necessary. 


252  HEAT 

PROBLEMS 

1.  The  specific  heat  of  water  is  much  greater  than  that  of  rocks  and  soils. 
How  does  this  in  part  account  for  the  fact  that  the  change  of  temperature  of 
the  land  between  day  and  night  and  between  winter  and  summer  is  much 
greater  than  that  of  the  ocean? 

2.  Are  equal  quantities  of  heat  required  to  raise  equal  volumes  of  different 
substances  through  equal  changes  of  temperature?     (Consult  table  of  densi- 
ties and  table  of  specific  heats.) 

3.  What  effect  has  the  large  specific  heat  of  water  on  the  sensation 
caused  by  putting  the  hand  in  hot  or  cold  water?     In  general,  how  does  the 
specific  heat  of  a  substance  affect  the  sensation  of  heat  or  cold  caused  by  it 
when  touched  (see  Art.  190)? 

4.  Dry  air  at  the  temperature  of  boiling  water  does  not  cause  a  burn, 
and  is  not  even  painfully  hot.     Why  not? 

6.  How  many  cubic  feet  of  air  can  be  warmed  one  degree  by  the  heat 
given  out  by  a  cubic  foot  of  water  in  cooling  one  degree? 

6.  Of  what  advantage  is  the  high  specific  heat  of  water  in  the  hot- water 
system  of  heating  buildings? 

7.  A  roll  of  lead  weighing  800  g.  is  heated  to  100°  and  placed  in  a  brass 
calorimeter  weighing  90  g.  and  containing  406.3  g.  of  water  at  16.2°.     The 
final  temperature  is  21°.      Find  the  specific  heat  of  lead. 

8.  A  kilogram  of  mercury  at  200°  and  a  kilogram  of  water  at  o°  are 
mixed.     Find  the  resulting  temperature,  no  allowance  being  made  for  the 
vessel. 

9.  A  piece  of  aluminum  weighing  60  g.  is  heated  to  63°,  and  placed  in  a 
copper  calorimeter  weighing  50  g.  and  containing  103  g.  of  alcohol  at  8°. 
The  temperature  of  the  alcohol  rises  to  17°.     Find  its  specific  heat,  taking 
the  specific  heat  of  copper  and  aluminum  from  the  table. 

VII.   FUSION  AND  SOLIDIFICATION 

213.  Change  of  State.  —  Among  the  various  effects  of 
heat  none  are  more  familiar  than  change  of  state.  Solids 
become  liquids  and  liquids  gases,  when  heat  is  received  in 
sufficient  quantity;  and  the  opposite  changes  of  state 
occur  when  heat  is  given  out.  Water  is  the  only  sub- 


FUSION  AND   SOLIDIFICATION  253 

stance  which  comes  under  ordinary  observation  in  all  three 
states.  Various  substances  are  familiar  both  in  the  solid 
and  the  liquid  states,  e.g.  glue,  wax,  jelly,  and  butter; 
others  in  the  liquid  and  the  gaseous  states  (the  volatile 
liquids  and  their  vapors),  e.g.  ether,  alcohol,  and  gasoline; 
and  still  others  only  as  solids  or  only  as  gases.  But  most 
substances  are  capable  of  existing  in  all  three  states,  at 
temperatures  ranging  very  high  in  some  cases  and  very 
low  in  others.  The  metals  melt  more  or  less  readily  in 
the  heat  of  a  furnace;  and  several  of  them,  including  iron, 
are  known  to  exist  as  vapors  in  the  atmosphere  of  the 
sun.  The  air  and  other  gases,  as  already  noted,  liquefy 
and  even  solidify  at  temperatures  approaching  absolute 
zero.  The  tissues  of  plants  and  animals  do  not  melt, 
but  undergo  chemical  change  at  high  temperatures,  by 
which 'they  are  broken  up  into  simpler  substances. 

In  studying  the  laws  and  principles  relating  to  change  of 
state,  water  is  taken  as  the  typical  example,  not  only  on 
account  of  its  convenience  and  familiarity,  but  principally 
because  the  melting,  freezing,  evaporation,  and  condensa- 
tion of  water  are  phenomena  of  the  greatest  importance 
in  nature. 

214.  Melting  of  Ice  and  Freezing  of  Water.  —  The  term 
"ice  cold"  seems  to  imply  that  the  temperature  of  ice  is 
always  the  same.  The  fact  is  that  ice  loses  heat  and  cools 
to  the  temperature  of  surrounding  bodies,  when  their 
temperature  is  below  zero;  and,  in  receiving  heat  at  all 
temperatures  below  zero,  it  becomes  warmer,  just  as  other 
substances  do.  The  specific  heat  of  ice  is  .504,  or  approxi- 
mately half  that  of  water  in  the  liquid  state ;  i.e.  a  gram  of 
ice  in  losing  .504  calories  falls  one  degree  in  temperature, 
and  in  receiving  .504  calories,  at  all  temperatures  below 


254  HEAT 

zero,  it  becomes  one  degree  wanner.  But  ice  can  not  be 
heated  above  o°  C.  When  ice  at  this  temperature  is  sur- 
rounded by  warmer  bodies,  even  when  thrown  into  boiling 
water  or  placed  on  a  hot  stove,  the  heat  received  does  not 
penetrate  the  ice,  and  only  causes  melting  at  its  surface. 
The  comparatively  slow  rate  at  which  ice  melts,  even  on 
a  hot  summer  day,  indicates  that  much  heat  is  required 
to  accomplish  the  change  of  state. 

Water  cools  as  it  loses  heat  until  its  temperature  falls 
to  o°.  With  further  loss  of  heat,  it  begins  to  freeze;  but  its 
temperature  remains  at  o°  until  it  is  all  frozen.  Ice  melts 
and  water  freezes  at  exactly  the  same  temperature,  but  under 
opposite  conditions  with  respect  to  the  transfer  of  heat. 
Ice  melts  only  in  proportion  to  the  heat  received  at  o°, 
and  water  freezes  only  in  proportion  to  the  heat  lost  at 
o°.  With  neither  a  gain  nor  a  loss  of  heat,  neither  melting 
nor  freezing  can  take  place. 

215.  Melting  Points.  —  Every  solid  that  can  be  melted 
has  a  constant  melting  point,  which  is  also  the  temperature 
at  which  it  freezes  or  solidifies.  Among  fusible  solids, 
some,  like  ice,  change  abruptly  from  the  solid  to  the  liquid 
state.  In  such  cases  the  melting  point  can  be  very  accu- 
rately determined.  Other  solids,  e.g.  sealing  wax,  glue, 
pitch,  and  glass,  gradually  soften  and  pass  by  continuous 
change  into  the  liquid  state.  In  such  cases  the  melting 
point  is  indefinite  just  to  the  extent  that  the  distinction 
between  the  solid  and  the  liquid  state  is  indefinite. 

TABLE  OF  MELTING  POINTS 

Alcohol    -130°  C.  Paraffin 54°  C. 

Mercury    -39        Beeswax    62 

Ice   o        Rose's  metal  (alloy  of  tin, 

Butter 33  lead,  and  bismuth)    96 


FUSION  AND  SOLIDIFICATION  255 

Sulphur -...'....  115°  C.  Aluminum    657°  C. 

Cane  sugar    170  Copper noo 

Solder,  soft     225  Glass    1000  to  1400 

Lead 327  Iron    1200  to  1600 

Zinc    420  Platinum 1775 

216.  Change  of  Volume  during  Fusion  and  Solidifica- 
tion. —  Most  substances  expand  in  melting  and  contract 
in  solidifying,  the  change  of  volume  in  some  cases  being 
considerable.  The  contraction  of  beeswax  or  paraffin  in 
solidifying  leaves  a  considerable  depression  at  the  center  of 
the  top  surface.  Metals,  with  few  exceptions,  also  contract 
in  solidifying.  Those  that  do  are  unsuitable  for  casting, 
as  they  take  only  an  imperfect  impression  of  the  mold. 
Cast  iron,  bismuth,  and  type  metal  (an  alloy  of  lead,  tin, 
and  antimony)  are  among  the  exceptions. 

Water  expands  in  solidifying,  the  increase  in  volume 
amounting  to  about  one  eleventh.  In  consequence  of 
this  expansion,  ice  floats  —  a  fact  of  great  importance  in 
nature.  If  water  contracted  in  freezing,  ice  forming  at 
the  surface  of  lakes  and  rivers  would  sink.  Freezing  would 
therefore  continue  rapidly  throughout  winter,  or  until  the 
lakes  and  rivers  were  frozen  solid ;  and  all  animal  life  inhab- 
iting them  would  be  destroyed. 

The  expansion  of  water  in  freez- 
ing is  responsible  for  the  bursting  of 
water-pipes  in  winter.  That  the  force 
of  expansion  is  practically  irresistible 
was  strikingly  shown  by  the  experi- 
ments of  Major  Williams,  in  Canada. 
"  Having  quite  filled  a  thirteen-inch 
bomb-shell  with  water,  he  firmly 
closed  the  touch-hole  with  an  iron 
plug  weighing  three  pounds,  and  ex- 
posed it  in  this  state  to  the  frost.  After  some  time  the  iron  plug 
was  forced  out  with  a  loud  explosion,  and  thrown  to  a  distance  of 


256  HEAT 

415  ft.,  and  a  cylinder  of  ice  8  in.  long  issued  from  the  opening. 
In  another  case  the  shell  burst  before  the  plug  was  driven  out,  and 
in  this  case  a  sheet  of  ice  spread  out  all  round  the  crack."  (Ganot.) 

"  Much  of  the  destruction  of  rocks  which  is  taking  place  on  the 
earth's  surface  is  due  to  the  same  quiet  but  intensely  powerful  action 
of  freezing  water.  Rain  sinks  into  the  cracks  and  pores  which  all 
rocks  are  liable  to  contain,  and  when  it  freezes  there,  the  crack  is 
inevitably  widened  or  the  structure  of  the  rock  loosened.  Thus  room 
is  made  for  more  water,  which  acts  in  the  same  way  when  it  freezes; 
and  so  by  degrees  immense  masses  of  rock  and  earth  are  loosened 
from  the  mountainside,  nor  does  the  action  end  until  the  material  is 
reduced  to  the  finest  soil."  (Madan.) 

Substances  that  expand  in  solidifying  have  a  crystalline  structure 
in  the  solid  state.  The  crystalline  structure  is  plainly  seen  in  the  ice 
that  first  forms  when  water  begins  to  freeze,  in  the  frost  that  gathers 
on  window  panes,  and  in  snow.  In  crystalline  solids  the  molecules 
are  arranged  in  clusters  of  definite  shape,  and  may  therefore  occupy 
a  greater  space  than  they  do  when  lying  loosely  side  by  side  in  the 
liquid  state,  just  as  a  number  of  bricks  occupy  more  space  when 
arranged  in  patterns  than  they  do  when  packed  in  layers. 

217.  Change  of  Melting  Point  due  to  Pressure.  —  Experiments 
have  shown  that  when  water  is  subjected  to  great  pressure  its  freezing 
point  is  lowered.  It  does  not  freeze  at  o°  because  the  expansion 
which  normally  accompanies  freezing  is  retarded.  In  agreement 
with  this  it  is  also  found  that  ice  melts  below  o°  under  great  pressure; 
for  the  pressure  tends  to  bring  about  the  decrease  of  volume  which 
accompanies  melting.  The  melting  and  freezing  points  are  equally 
lowered  by  a  given  pressure,  as  we  should  expect. 

Ice  has  been  melted  at  —  18°  under  a  pressure  estimated  at  several 
thousand  atmospheres.  The  change  in  the  melting  point  due  to  a 
pressure  of  one  atmosphere  is  only  .0072°,  and  would  escape  detection 
by  means  of  the  thermometers  used  in  elementary  physics;  yet  the 
effects  produced  under  certain  conditions  by  moderate  changes  of 
pressure  are  very  striking.  Thus  a  loop  of  fine  wire  to  which  weights 
are  attached  slowly  descends  through  a  block  of  ice  round  which  it 
has  been  passed  (Fig.  189) ;  yet,  after  it  has  passed  completely  through, 
the  ice  is  one  solid  piece  as  at  the  beginning.  The  pressure  of  the  wire 
very  slightly  lowers  the  melting  point  of  the  ice  immediately  beneath 


FUSION  AND   SOLIDIFICATION 


257 


FIG.  189.  —  Wire  Passing  through  Ice. 


it;  and  the  ice  melts,  receiving  the  necessary  heat  from  the  water 
just  above  the  wire.  This  water  freezes  in  losing  heat,  since  it  is 
relieved  from  the  pressure. 
The  process  is  continuous; 
for  the  water  from  the  melt- 
ing ice  below  the  wire  passes 
round  and  freezes  above  it. 
The  three  stages  of  the  pro- 
cess are  (i)  melting  under 
pressure,  (2)  change  of  posi- 
tion of  the  water,  (3) 
regelation  (refreezing)  under 
diminished  pressure. 

Snowballs  are  formed  by 
partial  melting  and  regela- 
tion of  the  snow  under  the 

pressure  of  the  hands.  The  slow  change  of  snow  into  the  clear  ice 
of  glaciers  is  due  to  the  same  action  under  gravity  pressure.  This 
action  continues  even  in  the  solid  ice  of  the  glacier,  which,  in  con- 
sequence, slowly  flows  down  the  mountain  valleys  at  a  rate  varying 
from  a  few  inches  to  one  or  two  feet  per  day. 

218.  Heat  of  Fusion.  —  Since  the  melting  of  ice  is  slow, 
even  in  warm  weather,  it  is  reasonable  to  conclude  that 
much  heat  is  required  to  bring  about  the  change  of  state. 
A  more  definite  idea  of  the  amount  of  heat  required  to  melt 
a  given  mass  of  ice  may  be  gained  by  applying  a  Bunsen 
flame  to  a  beaker  containing  a  mass  of  broken  ice,  and,  at 
the  same  time,  a  similar  flame  to  a  second  beaker  contain- 
ing an  equal  mass  of  water,  taken  at  the  temperature  of 
the  ice.  If  the  contents  of  the  first  beaker  are  constantly 
stirred  until  the  ice  is  all  melted,  the  temperature  of  the 
water  will  be  but  little  above  o°,  while  the  water  in  the  other 
beaker  will  be  found  nearly  boiling  hot.  The  main  point 
to  be  noted  is  that  the  change  of  state  in  the  one  case  and 
the  heating  in  the  other  are  accomplished  by  approxi- 
mately equal  quantities  of  heat. 


258  HEAT 

By  methods  adapted  to  accurate  measurement  it  has 
been  found  that  80  calories  are  required  to  convert  a  gram 
of  ice  at  o°  into  water  at  the  same  temperature.  When 
the  opposite  change  of  state  occurs,  an  equal  quantity*  of 
heat  is  given  out,  i.e.  a  gram  of  water  at  o°  freezes  only 
on  losing  80  calories.  This  quantity  is  called  the  heat  of 
fusion  of  ice.  The  heat  of  fusion  of  any  substance  is  the 
number  of  calories  required  to  melt  one  grym  of  ity-a.ft.er  it 
has  reached  its  melting  point,  or  the  number  of  calories 
given  out  by  one  gram  of  the  substance  in  solidifying,  with- 
out a  change  of  temperature  in  either  case.  The  heat  of 
fusion  of  water  is  much  larger  than  that  of  most  substances. 

TABLE  or  HEATS  OF  FUSION 

Calories  Calories 

Ice   80.0  Tin    14.25 

Paraffin  35.1  Sulphur   9.37 

Zinc 28.1  Lead 5.86 

Iron 23  to  33          Mercury 2.83 

219.  Transformations  of  Energy  during  Fusion  and 
Solidification.  —  Since  the  heat  received  by  a  solid  while 
it  is  melting  does  not  affect  its  temperature,  we  conclude 
that  this  energy  has  ceased  to  exist  as  heat  in  producing 
the  change  of  state.  Melting  may  be  said  to  consist  in 
overcoming  the  cohesion  which  binds  the  molecules  of  a 
solid  together.  In  doing  this  internal  work,  heat  becomes 
molecular  potential  energy  —  the  energy  of  an  altered 
molecular  condition.  This  energy  is  recovered  as  heat 
during  the  opposite  change  of  state,  as  the  principle  of  the 
conservation  of  energy  would  lead  us  to  expect.  These 
transformations  of  molecular  energy  may  be  illustrated 
by  means  of  two  balls  connected  by  a  rubber  band,  the  balls 
representing  molecules  and  the  rubber  band,  cohesion. 
In  pulling  the  balls  apart  work  is  done  against  the  force 


FUSION  AND   SOLIDIFICATION  259 

which  tends  to  hold  them  together.  This  work  is  stored 
as  potential  energy,  and  is  recovered  when  the  balls  are 
permitted  to  come  together  again. 

According  to  the  caloric  theory,  heat  always  remains  heat,  being 
(as  was  supposed)  a  form  of  matter.  In  the  language  of  this  theory, 
the  heat  that  disappears  during  fusion  and  vaporization  was  called 
"latent,"  i.e.  inactive  or  hidden;  and  heat,  properly  so  called,  was 
known  as  "sensible"  heat.  The  trio  of  misnomers  "sensible  heat," 
"latent  heat,"  and  "radiant  heat"  are  only  now  falling  into  dis- 
repute, half  a  century  and  more  after  the  overthrow  of  the  theory 
that  gave  rise  to  them. 

220.  Heat  of  Solution.  Freezing  Mixtures.  —  Work  is 
done  in  overcoming  cohesion  in  a  solid  when  it  is  dissolved 
as  well  as  when  it  is  melted ;  and  in  many  instances  there  is 
direct  experimental  evidence  that  heat  disappears  in  the 
process,  proving  that  this  work  is  accomplished  by  heat.* 
Thus  when  ammonium  chloride  or  ammonium  nitrate  is 
dissolved  in  water,  there  is  a  fall  of  temperature  of  several 
degrees ;  for  the  heat  required  to  dissolve  the  solid  is  taken 
from  the  nearest  available  source,  i.e.  the  water.  Solution 
differs  from  fusion  in  that  it  can  take  place  within  a  wide 
range  of  temperatures;  hence  the  temperature  continues  to 
fall  (unless  heat  is  received  from  outside  sources)  until 
all  the  solid  is  dissolved  or  until  the  solution  is  saturated. 

A  mixture  of  one  or  more  solids  and  a  liquid,  or  of  two 
solids,  is  called  a  freezing  mixture  if  the  solution  or  the 
liquefaction  of  the  solids  causes  a  fall  of  temperature  below 
zero.  The  following  are  examples  of  freezing  mixtures: 

i.  One  part  by  weight  of  ammonium  chloride  and  one  of  potassium 
nitrate  or  ammonium  nitrate,  powdered  together  and  dissolved  in 
two  parts  of  water.  Fall  of  temperature  about  20°. 

*  When  chemical  action  accompanies  solution,  it  may  result  in  a  rise  of 
temperature,  the  heat  generated  by  the  chemical  action  being  greater,  in  such 
cases,  than  the  heat  lost  in  solution. 


260  HEAT 

2.  One  part  of  table  salt  and  two  parts  of  snow  or  crushed  ice. 
Fall  of  temperature  to  about  — 18°.     The  strong  attraction  of  salt 
for  water  causes  the  ice  to  melt  rapidly,  and  at  a  temperature  below 
its  normal  melting  point.     The  heat  required  to  melt  the  ice  and  to 
dissolve  the  salt  is  taken  first  from  the  ice  and  salt,  then,  by  conduc- 
tion, from  surrounding  bodies.     This  freezing  mixture  is  well  known 
from  its  use  in  making  ice  cream. 

3.  One  part  each  of  crystallized  calcium  chloride  and  snow  or 
crushed  ice.     Fall  of  temperature  to  about  —  40°. 

PROBLEMS 

1.  What  determines  whether,  in  a  mixture  of  ice  and  water,  both  at  o°, 
the  ice  will  melt  or  the  water  freeze? 

2.  The  heat  of  fusion  of  iron  being  much  less  than  that  of  ice,  how  does 
it  happen  that  iron  does  not  readily  melt  and  ice  does? 

3.  What  purpose  is  served  by  vessels  of  water  placed  in  a  cellar  where 
vegetables  are  stored  or  in  a  greenhouse  on  a  frosty  night? 

4.  (a)  Do  freezing  and  thawing  take  place  more  or  less  rapidly  than  they 
would  if  the  heat  of  fusion  of  ice  were  less?     (&)  Of  what  importance  is  this 
in  the  economy  of  nature? 

5.  How  much  heat  is  required  to  convert  750  g.  of  ice,  taken  at  -20°, 
into  water  at  50°? 

6.  How  many  grams  of  ice  at  o°  can  be  melted  by  500  g.  of  water  at  6o°? 

7.  A  piece  of  aluminum  weighing  250  g.  and  heated  to  100°  is  placed  in  a 
dry  cavity  in  a  block  of  ice,  and  melts  68,8  g.  of  the  ice.     Find  the  specific 
heat  of  the  aluminum,  taking  the  heat  of  fusion  of  ice  as  80  calories. 

8.  The  quantity  of  heat  that  melts  one  gram  of  ice,  taken  at  o°,  would 
melt  how  many  grams  of  lead,  also  taken  at  o°? 

9.  Does  ice  mixed  with  salt  melt  more  or  less  rapidly  than  ice  alone? 
(Try  the  experiment.)     Account  for  the  result.     If  the  temperature  of  the 
freezing  mixture  is  -18°,  do  we  infer  that  the  ice  is  melting  at  this  tempera- 
ture? 

10.  Does  ice  in  a  refrigerator  serve  its  purpose  by  its  mere  presence  or 
by  melting?     Explain. 

VIII.    VAPORIZATION  AND  CONDENSATION 

221.   Vaporization.  —  The  gaseous  form  of  a  substance 
that  exists  as  a  liquid  or  a  solid  at  ordinary  temperatures 


VAPORIZATION  AND   CONDENSATION  261 

is  called  a  vapor,  and  the  change  of  a  liquid  or  a  solid  to 
the  gaseous  state  is  called  vaporization.  Vaporization  may 
take  place  at  the  free  surface  of  a  liquid  or  within  its  mass. 
In  the  first  case  it  is  generally  called  evaporation;  in  the 
second  case,  boiling.  A  volatile  liquid  is  one  that  evapo- 
rates readily,  e.g.  gasoline,  alcohol,  and  ether. 

Liquids  boil  at  definite  temperatures,  for  reasons  which 
are  considered  later;  they  evaporate  at  all  temperatures, 
but  more  rapidly  as  the  temperature  rises.  A  damp  cloth 
dries  slowly  in  a  cold  room,  more  quickly  in  the  warm  sun- 
shine, and  very  quickly  before  a  hot  fire. 

Evaporation  is  due  to  molecular  motion.  Some  of  the 
molecules  of  a  liquid,  in  their  irregular  and  unequal  motion, 
reach  the  surface  with  a  sufficient  upward  velocity  to  carry 
them  into  the  space  above,  out  of  the  range  of  cohesion, 
where  they  exist  as  a  gas  or  vapor.  With  a  rise  of  tempera- 
ture the  velocity  of  the  molecules  is  increased,  and  more  of 
them  are  able  to  escape  from  the  liquid  in  a  given  time. 

Evaporation  takes  place  on  the  largest  scale  from  the 
surface  of  the  oceans,  lakes,  ponds,  and  streams,  and  from 
damp  soil ;  in  consequence  of  which  the  air  always  contains 
a  greater  or  less  amount  of  water  vapor. 

222.  Saturated  Vapor.  —  A  liquid  kept  in  an  open  vessel 
continues  to  evaporate  until,  in  the  course  of  time,  it  en- 
tirely disappears.  In  a  closed  vessel  this  does  not  happen. 
A  small  quantity  of  such  a  volatile  liquid  as  ether  can  be 
kept  indefinitely  in  a  large  bottle,  if  only  it  is  tightly  stop- 
pered. Evaporation  takes  place  for  a  time  into  the  closed 
space  above  the  liquid,  and  then  apparently  ceases.  The 
vapor  is  then  as  dense  as  it  will  become  at  the  existing 
temperature,  however  much  or  little  of  the  liquid  may  still 
remain,  and  however  long  it  may  stand.  Any  vapor  in 


262  HEAT 

this  condition  is  called  a  saturated  vapor.  When  the  con- 
dition of  a  vapor  is  such  that  further  evaporation  of  the 
liquid  into  the  same  space  is  possible,  it  is  said  to  be  unsat- 
urated  or  superheated. 

Saturation  is  explained  by  the  kinetic  theory  of  matter 
as  follows :  Whenever  a  liquid  and  its  vapor  are  in  contact, 
there  is  a  constant  exchange  of  molecules  between  them. 
Molecules  of  the  liquid,  breaking  away  from  the  surface, 
become  a  part  of  the  vapor;  and  molecules  of  the  vapor, 
striking  the  surface  of  the  liquid,  are  captured  and  become 
a  part  of  the  liquid  again.  As  evaporation  into  a  closed 
space  continues,  the  density  of  the  vapor  increases,  and  an 
increasing  number  of  its  molecules  return  to  the  liquid 
in  a  given  time.  Finally,  the  condition  is  reached  in  which 
there  is  an  equal  exchange  of  molecules  between  the  liquid 
and  its  vapor.  The  two  are  then  in  equilibrium  with  each 
other,  and  the  vapor  is  saturated. 

223.  Vapor  Pressure.  —  A  vapor,  like  any  gas,  exerts 
a  certain  pressure  which,  at  a  constant  temperature,  is 
proportional  to  its  density;  but  the  behavior  of  vapors 
differs  from  that  of  other  gases  in  important  respects,  as 
shown  by  the  following  experiment. 

A  simple  barometer  is  set  up,  and  a  drop  or  two  of  ether 
introduced  into  the  bottom  of  it,  by  means  of  a  curved 
dropping  tube,  care  being  taken  to  let  no  air  enter  (Fig. 
190).  The  ether  evaporates  as  it  rises  through  the  column 
of  mercury,  and  the  pressure  that  it  exerts  as  a  vapor 
causes  a  depression  of  the  mercury  column.  When  more 
ether  is  introduced,  it  rises  without  evaporating  and 
remains  as  a  liquid  above  the  mercury.  The  space  above 
the  liquid  is  now  filled  with  its  saturated  vapor.  The  pres- 
sure exerted  by  the  vapor,  expressed  in  centimeters  of 


VAPORIZATION  AND   CONDENSATION  263 

mercury,  is  measured  by  the  difference  between  the  present 
height  of  the  mercury  column  and  its  height  before  the 
ether  was  introduced.  When  the  tube  is  inclined,  the  space 
occupied  by  the  vapor  becomes  smaller;  but  the  vertical 
height  of  the  mercury  column  remains  the  same  as  before, 
showing  that  the  vapor  pressure  is  unchanged.  This 
holds  true  as  the  tube  is  further  inclined,  until  the  space 
above  the  liquid  entirely  disappears  (provided 
all  air  has  been  excluded  from  the  tube).  In 
inclining  the  tube  the  vapor  is  evidently  not 
compressed  and  made  denser;  for  in  that  case 
it  would  exert  an  increased  pressure.  The  fact 
is  that  the  vapor  condenses  as  rapidly  as  its 
volume  is  diminished,  and  the  density  of  the 
remainder  is  unchanged.  This  behavior  is  char- 
acteristic of  all  saturated  vapors.  The  density 
and  pressure  of  a  saturated  vapor  are  both  at  a 
maximum  for  the  existing  temperature.  Any  at- 
tempt at  compression  only  results  in  condensa- 
tion, which  takes  place  in  exact  proportion  to 
the  decrease  of  volume.  (Contrast  this  behavior 

with  that   of  "perfect"  gases,  as  expressed  in 

^     ,  ,  FIG.  190. 

Boyle  s  law.) 

When  the  tube  in  the  above  experiment  is  returned  to 
the  vertical  position  and  the  ether  warmed  by  clasping 
the  tube  in  the  hands,  the  mercury  descends  further,  show- 
ing an  increase  of  vapor  pressure  with  a  rise  of  temperature. 
This  is  partly  due  to  the  heating  of  the  vapor  already  in 
the  tube,  but  chiefly  to  the  evaporation  of  more  ether. 
The  density  and  pressure  of  a  saturated  vapor  increase  with 
the  temperature.  If  there  were  no  more  ether  in  the  tube 
to  evaporate,  heat  would  cause  expansion  of  the  existing 
vapor,  and  it  would  become  less  dense,  and  unsaturated. 


264  HEAT 

Similar  results  are  obtained  throughout  when  alcohol 
is  substituted  for  ether  in  the  experiment;  but  they  are  all 
on  a  greatly  reduced  scale,  for  the  maximum  pressure  of 
alcohol  vapor  is  much  less  than  that  of  ether  at  the  same 
temperature.  With  water  the  results  are  very  slight. 
At  20°  the  maximum  vapor  pressure  of  ether  is  43.28  cm. 
(of  mercury),  that  of  alcohol  is  4.45  cm.,  and  that  of  water 
1.74  cm.  At  the  same  temperature  the  maximum  pressures 
of  different  vapors  are  unequal. 

The  behavior  of  unsaturated  or  superheated  vapors  is 
approximately  like  that  of  perfect  gases,  as  expressed  in 
the  laws  of  Boyle,  Gay-Lussac,  and  Charles. 

224.  Mixtures  of  Gases  and  Vapors.  —  In  a  vacuum 
evaporation  is  very  rapid,  and  the  space  is  almost  immedi- 
ately filled  with  the  saturated  vapor  of  the  liquid.     In  the 
presence  of  air  or  any  other  gas,  evaporation  takes  place 
much  more  slowly;  but  it  does  not  cease  until  any  inclosed 
space  above  the  liquid  contains  as  much  of  the  vapor  as 
it  would  if  the  other  gas  were  not  present,     (i)  The  quantity 
of  vapor  which  saturates  a  given  space  is  the  same,  at  the 
same  temperature,  whether  this  space  contains  a  gas  or  is  a 
vacuum.     (2)   The  pressure  exerted  by  a  mixture  of  one  or 
more  gases  and  vapors  is  equal  to  the  sum  of  the  pressures 
which  each  would  exert  if  it  occupied  the  same  space  alone. 
The  most  familiar  example  of  such  a  mixture  is  the  atmos- 
phere.    The  statements  in  italics  are  known  as  Dalton's 
laws. 

225.  Loss  of  Heat  in  Evaporation.  —  Common  observa- 
tion teaches  that  evaporation  is  a  cooling  process.     The 
skin  is  cooled  by  the  evaporation  of  water  or  perspiration 
from  it.     This  is  especially  noticeable  in  a  draft,  which 
causes   more   rapid   evaporation   by   carrying   the   vapor 


VAPORIZATION  AND   CONDENSATION  265 

away  as  fast  as  it  is  formed.  Damp  earth  is  considerably 
cooler  than  dry  earth  on  a  dry,  hot  day,  when  evaporation 
is  rapid.  The  very  rapid  evaporation  of  ether  and  other 
highly  volatile  liquids  causes  much  greater  cooling. 

The  heat  lost  during  evaporation  is  required  to  produce 
the  change  of  state,  just  as  heat  is  required  to  melt  or 
dissolve  a  solid.  In  evaporation,  as  in  solution,  this  heat 
is  taken  from  the  nearest  available  source  —  first  the  liquid 
itself,  then  adjacent  bodies.  The  nature  of  the  work  done 
by  heat  during  vaporization  is  discussed  in  Art.  234. 

226.   Conditions  Affecting  the  Rate  of  Evaporation.  — 

The  rate  of  evaporation  of  a  liquid  is  affected  by  various 
conditions,  as  follows: 

Temperature.  —  The  rate  of  evaporation  increases  with 
a  rise  of  temperature  (Art.  221). 

Density  of  the  Vapor.  —  The  evaporation  of  a  liquid 
decreases  as  the  space  about  it  approaches  saturation  by 
its  own  vapor,  and  ceases  when  that  space  is  saturated 
(Art.  222). 

Presence  of  Air  or  other  Gas.  —  The  rate  of  evaporation 
increases  as  the  density  of  the  air  or  other  gas  surrounding 
the  liquid  is  diminished.  It  takes  place  most  rapidly  in 
a  vacuum  (Art.  224).  This  may  be  readily  shown  as 
follows:  A  small  beaker  is  partly  filled  with  ether  and  left 
exposed  to  the  air  of  the  room;  and  a  second  beaker  con- 
taining ether  is  placed  under  the  receiver  of  an  air  pump. 
The  ether  in  the  exhausted  receiver  evaporates  so  rapidly 
that  its  temperature  falls  several  degrees  below  zero  in  a 
few  minutes,  while  the  ether  in  the  other  beaker  is  cooled 
comparatively  little  by  the  much  slower  evaporation  into 
the  air.  The  beaker  under  the  receiver  should  stand  on 
wood  or  cork  to  prevent  conduction  from  the  metal  plate 


266  HEAT 

of  the  pump.     If  the  support  of  the  beaker  is  wet  with  a 
few  drops  of  water,  the  beaker  will  be  frozen  to  it. 

Changes  of  barometric  pressure  are  not  sufficiently 
great  to  affect  the  rate  of  evaporation  in  the  open  air  to 
any  appreciable  extent. 

Air  Currents.  —  The  rate  of  evaporation  in  air  in- 
creases with  a  more  rapid  change  of  air  about  the  liquid. 
Currents  of  air  carry  the  vapor  away  from  the  space 
about  the  liquid;  and  the  stronger  the  currents  are  the 
farther  will  this  space  be  from  saturation.  It  is  a 
familiar  fact  that  moisture  quickly  disappears  in  a  dry, 
hot  wind. 

Area  of  Free  Surface.  —  Evaporation  increases  with  an 
increase  of  the  free  surface  of  the  liquid.     Water  evapo- 
rates slowly  from  a  cup,  more  rapidly  from  a  broad  and 
shallow  dish,  and  still  more  rapidly  from  wet  clothes  hung 
on  a  line.     A  small  quantity  of  ether  in  a  beaker  is  quickly 
cooled  below  zero  by  a  current  of  air  forced 
through  it  from  a  small  bellows  (Fig.  191). 
The  ether  evaporates  into  the  bubbles  of  air 
as  they  rise  through  the  liquid.     The  area 
of   the  evaporating  surface  is   thus  greatly 
increased;  and  there  is  a  constant  renewal  of  unsaturated 
space,  into  which  evaporation  can  take  place. 

Nature  of  the  Liquid.  —  Under  the  same  conditions  the 
rate  of  evaporation  differs  with  different  liquids. 

227.  Water  Vapor  in  the  Atmosphere.  —  The  atmos- 
phere is  a  mixture  of  several  gases,  principally  nitrogen 
and  oxygen.  The  only  other  constituents  of  importance 
are  carbon  dioxide  and  water  vapor.  All  of  the  constitu- 
ents of  the  atmosphere  except  water  vapor  are  practically 
constant  in  amount;  the  water  vapor  varies  from  an  inap- 


VAPORIZATION  AND   CONDENSATION  267 

preciable  fraction  to  about  2  %  of  the  whole,  the  average 
amount  being  not  far  from  i  %. 

The  condition  of  the  water  vapor  in  the  air  with  respect 
to  saturation  is  not  in  the  least  affected  by  the  presence 
of  the  other  gases  (Dal ton's  first  law),  and  depends  only 
upon  its  own  density  and  temperature  (which,  of  course, 
is  the  temperature  of  the  air) ;  yet  common  forms  of  expres- 
sion seem  to  imply  that  the  presence  and  condition  of  the 
vapor  are  due  to  some  action  of  the  air.  Thus  when  the 
water  vapor  in  the  air  is  saturated,  we  say  that  the  air 
is  saturated  or  that  the  air  has  all  the  moisture  it  can  hold; 
although,  strictly  speaking,  it  is  the  space  that  has  all  the 
water  vapor  it  can  hold  at  the  existing  temperature.  There 
is  perhaps  no  objection  to  the  use  of  such  expressions  when 
their  true  meaning  is  understood. 

The  air  is  generally  not  saturated;  it  is  evidently  not 
saturated  whenever  further  evaporation  can  take  place. 
Unsaturated  air  may  become  saturated  (i)  by  further 
evaporation,  (2)  by  a  fall  of  temperature,  or  (3)  by  the  two 
processes  combined.  Saturation  results  from  a  sufficient 
fall  of  temperature  because  the  density  of  a  saturated  vapor 
is  less  at  lower  temperatures  (Art.  223,  third  paragraph). 
Consequently  when  the  quantity  of  water  vapor  in  the  air 
is  less  than  that  required  for  saturation  at  the  existing 
temperature,  it  is  sufficient  to  cause  saturation  at  a  definite 
lower  temperature,  called  the  dew-point. 

228.  The  Dew-point.  —  The  temperature  at  which  the 
water  vapor  present  in  the  air  at  any  time  would  be  satu- 
ratecl  is  called  the  dew-pnint  of  fbg  air  a,t  that  H™**  When 
any  body  of  air  is  cooled  to  its  dew-point,  condensation  of 
water  vapor  begins,  and  it  continues  as  long  as  the  tempera- 
ture continues  to  fall.  The  moisture  that  gathers  on  the 


268  HEAT 

outside  of  a  pitcher  of  ice- water  is  a  familiar  example. 
This  moisture  comes  from  the  surrounding  air,  which  is 
cooled  by  contact  with  the  pitcher.  The  temperature 
usually  falls  several  degrees  below  the  dew-point,  causing 
a  considerable  deposit  which  runs  down  the  sides.  (What 
error  is  implied  in  calling  this  phenomenon  "sweating"?) 

The  dew-point  can  be  determined  experimentally  by 
slowly  cooling  the  contents  of  a  vessel  till  the  first  trace 
of  moisture  appears  on  its  surface.  The  temperature  of 
the  vessel  and  contents  when  this  occurs  is  the  dew-point 
of  the  surrounding  air.  The  vessel  should  be  one  upon 
which  a  thin  film  of  moisture  can  easily  be  seen,  such  as  a 
nickel-plated  calorimeter.  It  can  be  cooled  with  water 
to  which  ice  or  ammonium  chloride  is  added,  or  ice  and 
salt  if  the  dew-point  is  below  zero,  or  with  ether,  cooled 
by  evaporation. 

.The  dew-point  varies  between  wide  limits.  In  winter 
it  is  often  many  degrees  below  zero.  It  is  always  consider- 
ably below  the  temperature  of  the  air  when  the  air  is  not 
noticeably  damp,  and  is  as  high  as  the  temperature  of  the 
air  only  when  the  air  is  saturated. 

229.  Humidity.  —  The  ratio  of  the  amount  of  water 
vapor  present  in  the  air  at  any  time  to  the  whole  amount 
that  it  would  contain  if  saturated  at  the  existing  tempera- 
tur*e  is  called  the  relative  humidity  or,  simply,  the  humidity 
of  the  air  at  the  time.  This  ratio  is  usually  expressed  as 
a  percentage.  Thus  the  humidity  is  75  %  when  the  air 
contains  three  fourths  as  much  water  vapor  as  it  would 
if  it  were  saturated  at  the  same  temperature.  Humidity 
is  simply  the  measure  of  the  dampness  of  the  air.  It  is 
high  when  the  air  is  damp  and  low  when  it  is  dry. 

The  humidity  of  the  air  varies  not  only  with  the  amount 


VAPORIZATION  AND   CONDENSATION  269 

of  water  vapor  in  it,  but  also  with  its  temperature.  As  air 
cools  the  humidity  rises,  until  at  the  dew-point  it  is  100  %. 
Conversely  a  rise  of  temperature  lowers  the  humidity; 
for  the  quantity  of  vapor  actually  present  is  not  changed, 
while  the  capacity  of  the  air  for  vapor  is  increased.  Thus 
damp  air  in  a  cold  room  is  dried  by  heating  it,  although 
there  is  no  less  water  vapor  in  the  room  after  the  heating 
than  there  was  before. 

The  temperature,  pressure,  and  humidity  of  the  air  and 
the  direction  and  velocity  of  the  wind  are  the  principal 
atmospheric  conditions  which  determine  the  weather. 
These  conditions  are  regularly  measured  and  recorded  at 
all  stations  of  the  Weather  Bureau,  and  the  weather  fore- 
cast is  based  upon  them.  The  humidity  can  be  determined 
in  various  ways,  by  means  of  instruments  called  hygrom- 
eters (Greek  hygros,  moist,  and  metron,  measure).  A 
dew-point  hygrometer  is  merely  an  instrument  for  the  con- 
venient determination  of  the  dew-point.  Knowing  the 
temperature  of  the  air  and  the  dew-point,  the  humidity 
is  computed  with  the  aid  of  a  table  of  densities  of  satu- 
rated water  vapor.  Suppose,  for  example,  that  the  tem- 
perature is  30°  and  the  dew-point  20°.  There  is  then 
enough  moisture  in  the  air  to  saturate  it  at  20°;  and  it  is 
found  from  the  table  that  a  cubic  meter  of  saturated  air  at 
this  temperature  contains  14.3  g.  of  water  vapor,  while,  if 
saturated  at  30°,  it  would  contain  26.2  g.  The  humidity 
is  therefore  14.3  -f-  26.2  or  55%  nearly.  i 

The  wet-and-dry-bulb  hygrometer  -(Fig.  192)  is  more  convenient, 
and  hence  is  more  generally  used.  It  consists  of  two  thermometers, 
the  bulb  of  one  of  which  is  kept  moist  by  means  of  a  cotton  wick 
surrounding  it  and  dipping  into  a  vessel  of  water.  The  constant 
evaporation  about  the  bulb  of  this  thermometer  lowers  its  tempera- 
ture more  or  less,  according  to  the  rate  of  evaporation;  and  this 


270 


HEAT 


depends  upon  the  humidity  and  temperature  of  the  air.  In  saturated 
air  the  readings  of  the  thermometers  are  equal.  (Why?)  Before 
reading  the  wet-bulb  thermometer,  it  is  whirled  rapidly  through  the 
air,  or  a  current  of  air  is  driven  over  the  bulb.  (Why?)  The  hu- 
midity corresponding  to  the  observed  temperatures  of  the  wet  and 
the  dry  bulbs  is  found  directly  from  a  table. 

The  greater  or  less  humidity  of  the  air  affects  our  bodily 
comfort  through  its  influence  on  the  evaporation  of  mois- 
ture from  the  skin  and  clothing.  Generally  speaking,  a 
medium  humidity  is  most  agreeable,  for  the  skin  is  then 
neither  too  moist  nor  too  dry.  A  very  damp  atmosphere 
adds  greatly  to  the  discomfort  of  cold  weather,  and  in  hot 
weather  it  endangers  health  and  even  life  itself.  The  heat 
of  the  body  is  generated  by  chemical  action, 
which  is  constantly  going  on  within  it.  In 
winter  this  heat  is  retained  by  heavy  clothing 
of  poorly  conducting  materials;  in  summer 
we  wear  light  garments  of  cotton  and  linen 
to  permit  its  ready  escape.  But  on  the 
hottest  days  this  alone  is  insufficient.  In- 
deed, when  the  temperature  of  the  air  is 
equal  to  or  higher  than  that  of  the  body 
(98°  F.),  loss  of  heat  by  conduction  ceases, 


FIG.  1 92.—  and  nature  provides  a  substitute  in  abundant 

Hygrometer.  .  .  ,      f 

perspiration,  especially  during  active  exercise, 
when  the  generation  of  heat  in  the  body  is  most  rapid. 
But  perspiration  is  of  no  avail  unless  it  evaporates, 
for  the  cooling  effect  is  due  to  the  heat  taken  from  the 
body  in  evaporation.  Hence  hot  weather  is  especially 
oppressive  and  dangerous  when  evaporation  is  retarded 
by  excessive  humidity.  In  the  very  dry  atmosphere  of 
deserts  there  is  comparatively  little  danger  of  sunstroke 
even  at  temperatures  above  100°  F. 


VAPORIZATION  AND   CONDENSATION  271 

230.   Condensation  of  Water  Vapor  in  the  Atmosphere. 

-  Water  vapor  is  always  invisible.  The  visible  forms  of 
moisture  in  the  atmosphere,  such  as  fog,  clouds,  and  the 
so-called  " steam"  near  the  spout  of  a  kettle  in  which 
water  is  boiling,  consist  of  minute  particles  of  liquid  water, 
due  to  the  condensation  that  accompanies  a  fall  of  tempera- 
ture after  the  dew-point  is  reached.  Dew,  frost,  rain, 
sleet,  hail,  and  snow  are  the  various  forms  in  which  the 
water  vapor  in  the  air  is  condensed  and  precipitated. 

Dew  is  formed  by  condensation  of  vapor  from  the  air  immediately 
surrounding  the  bodies  on  which  it  appears.  This  occurs  when  air 
that  is  saturated,  or  nearly  so,  is  cooled  below  the  dew-point  by  con- 
tact with  colder  objects.  Dew  forms  most  abundantly  on  the  coldest 
objects,  which  are  in  general  the  best  radiators  and  the  poorest  con- 
ductors; for  such  objects  lose  heat  by  radiation  more  rapidly  than  they 
receive  it  from  the  earth  by  conduction,  until  they  become  several 
degrees  colder  than  the  air.  Grass,  leaves,  and  boards  are  good  ex- 
amples. Dew  forms  only  at  night,  and  most  abundantly  toward 
morning,  when,  by  cooling,  the  air  has  become  nearly  saturated.  It 
forms  only  on  calm,  clear  nights;  for  on  clear  nights  cooling  is  most 
rapid  (Art.  200),  and  it  is  only  on  calm  nights  that  any  portion  of 
the  air  remains  long  enough  in  contact  with  cold  surfaces  to  be  cooled 
to  the  dew-point. 

Frost.  —  When  the  dew-point  is  below  zero,  condensation  takes 
place  in  the  form  of  frost,  under  conditions  otherwise  the  same  as 
are  necessary  for  the  formation  of  dew.  The  water  vapor  then  crys- 
tallizes in  the  solid  state  as  it  condenses,  without  passing  through  the 
intermediate  state  of  a  liquid. 

Clouds  and  Fog.  —  The  condensation  of  vapor  in  the  air  near  the 
earth  produces  a  fog;  at  higher  altitudes  it  forms  a  cloud.  A  fog 
consists  of  minute  globules  of  liquid  water.  A  cloud  is  composed 
of  liquid  particles,  like  a  fog,  or  of  ice  particles,  depending  upon  the 
temperature.  The  cooling  of  the  air  by  which  clouds  are  formed  may 
be  brought  about  by  radiation,  by  the  contact  and  partial  mixing 
of  a  vapor-laden  current  with  a  current  of  colder  air,  or  by  the  expan- 
sion of  ascending  currents.  The  last-named  process  requires  a  few 
words  of  explanation.  Gases  are  always  cooled  by  expansion  (Art. 


272  HEAT 

235);  and  ascending  currents  of  air  expand,  in  consequence  of  the 
diminished  pressure  at  higher  altitudes.  When  moist  air  is  thus 
cooled  below  the  dew-point,  towering  masses  of  cloud,  known  as 
"  thunder-heads,"  are  formed.  The  cooling  of  air  by  expansion,  and 
the  resulting  condensation  of  vapor,  are  beautifully  shown  in  the 
following  experiment.  A  large  flask  containing  a  little  warm  water 
is  tightly  closed  with  a  rubber  stopper,  and  vigorously  shaken  to 
saturate  the  inclosed  air.  It  is  then  placed  under  the  receiver  of  an 
air  pump.  On  exhausting  the  receiver,  the  stopper  is  driven  out  by 
the  pressure  within  the  flask;  and  the  flask  is  instantly  filled  with 
a  dense  fog,  formed  by  the  sudden  expansion  and  cooling  of  the 
saturated  air. 

Rain,  Sleet,  Snow,  and  Hail.  —  As  the  particles  of  a  cloud  grow 
by  further  condensation  or  by  uniting  with  one  another,  they  may 
become  too  large  to  be  sustained  in  the  air,  and  they  then  fall  as  rain. 
When  rain  freezes  in  falling  through  a  layer  of  colder  air,  it  is  called 
sleet.  Snow  is  formed  by  the  condensation  of  vapor  in  the  atmosphere 
at  temperatures  below  zero.  The  vapor  passes  directly  into  the  solid 
state,  as  in  the  formation  of  frost.  Hailstones  are  masses  of  ice,  or 
of  ice  and  snow,  which  are  sometimes  an  inch  or  two  in  diameter. 
They  are  often  made  up  of  several  layers  or  shells  of  ice  and  snow, 
showing  that  they  have  passed  through  a  variety  of  atmospheric 
conditions.  Hailstones  of  large  size  are  formed  only  in  violent 
storms;  but  the  exact  manner  of  their  formation  is  not  known. 

PROBLEMS 

1.  Give  two  reasons  why  a  liquid  evaporates  more  rapidly  in  a  wide  and 
shallow  vessel  than  it  does  in  an  unstoppered  bottle. 

2.  (a)  Why  does  the  breath  often  form  a  visible  cloud  on  a  cold  day? 
(b)  Is  it  more  likely  to  do  so  when  the  humidity  of  the  air  is  high  or  low? 

3.  The  moisture  from  the  spout  of  a  kettle  of  boiling  water  is  invisible 
for  a  few  inches  beyond  the  spout  then  for  some  distance  farther  it  forms  a 
cloud;  still  farther  it  is  all  invisible  again.     Account  for  these  facts. 

4.  Why  does  frost  form  on  the  inside  of  a  window-pane  but  not  on  the 
outside? 

5.  Is  frost  frozen  dew? 

6.  Why  does  frost  form  on  board  walks  when  it  does  not  on  cement 
walks? 


VAPORIZATION  AND   CONDENSATION  273 

231.  Boiling.  —  When  fresh  water  is  heated,  dissolved 
air  is  given  off  in  the  form  of  minute  bubbles,  which  begin 
to  form  on  the  sides  and  bottom  of  the  vessel  as  soon  as 
the  water  has  become  slightly  warm.  These  bubbles 
often  rise  to  the  surface  in  large  numbers,  where  the  air 
that  they  contain  escapes.  After  the  water  has  become 
hot,  much  larger  bubbles  begin  to  form  at  the  bottom  where 
the  heat  is  applied.  These  are  bubbles  of  steam,  or  water 
vapor.  They  rise  rapidly,  but  disappear  before  reaching 
the  surface,  being  condensed  by  the  cooler  water  near  the 
top.  It  is  the  collapse  of  these  first  bubbles  of  steam  that 
causes  the  singing  of  a  kettle  of  water  shortly  before  it 
begins  to  boil.  As  the  temperature  of  the  water  approaches 
1 00°  C.,  the  bubbles  rise  higher,  until  finally  they  burst 
at  the  surface,  throwing  the  water  violently  about.  The 
water  is  then  boiling.  In  general,  any  liquid  is  said  to 
boil  when  bubbles  of  its  vapor  form  within  it,  rise  to  the 
surface,  and  break. 

The  temperature  of  a  liquid,  when  boiling  in  an  open 
vessel,  varies  only  slightly,  if  at  all,  however  slow  or  rapid 
the  boiling;  and  the  temperature  of  the  escaping  vapor  is 
constant.  Heat  energy  is  required  to  produce  the  change 
of  state,  and  a  more  abundant  supply  of  heat  merely  causes 
the  change  to  take  place  more  rapidly.  Boiling  is  in  this 
respect  like  melting. 

A  bubble  of  vapor  within  a  liquid  sustains  a  pressure 
which  is  made  up  of  the  gravity  pressure  of  the  liquid  and 
the  transmitted  pressure  of  the  air,  or  other  gas,  upon 
its  free  surface.  If  the  pressure  of  the  saturated  vapor  of 
the  liquid  is  less  than  this  at  the  existing  temperature, 
internal  vaporization  can  not  take  place,  and  if  bubbles 
of  the  vapor  are  present  they  can  not  withstand  the  pres- 
sure, and  are  immediately  condensed.  This  is  what  hap- 


274 


HEAT 


pens,  as  we  have  seen,  when  the  first  bubbles  of  steam  rise 
into  cooler  water  at  the  top,  shortly  before  water  begins 
to  boil.  At  small  depths  in  a  liquid  the  gravity  pressure 
is  small,  compared  with  the  pressure  of  the  air,  and  may 
be  disregarded.  The  boiling  point  of  a  liquid  is  therefore 
defined  as  the  temperature  at  which  the  pressure  of  its  satu- 
rated vapor  is  equal  to  the  pressure  upon  the  free  surface  of 
the  liquid,  a  pressure  of  one  atmosphere  being  understood 
unless  otherwise  stated. 


BOILING  POINTS  UNDER  A  PRESSURE  OF  ONE  ATMOSPHERE 

Ether     34.6°    Turpentine   160° 

Chloroform      61.2      Glycerin 290 

Alcohol     . . 78.4      Mercury 357 

Water     . . 100.        Sulphur 445 

232.   Effect  of  Pressure  upon  the  Boiling  Point.  —  An 

increase  of  pressure  upon  the  surface  of  a  liquid  raises  its 

boiling  point,  for  the  vapor 
bubbles  within  the  liquid  must 
exert  the  increased  pressure, 
and  this  is  possible  only  at 
a  higher  temperature.  Thus 
in  engine  boilers  the  tempera- 
ture of  the  boiling  water  and 
the  steam  steadily  rises  during 
the  process  of  "  getting  up 
steam."  When  the  steam  gage 
registers  a  pressure  of  150  Ib. 
(per  square  inch)  the  tempera- 
ture is  185°  C.  (See  table 

FIG.  193—  Franklin's  Experiment.    ^^       Conversely  a  decrease 

of    pressure    lowers   the   boiling   point.     This   is   readily 
shown   by  either   of   the  following  experiments,     (i)  An 


VAPORIZATION  AND   CONDENSATION 


275 


open  flask  containing  water  at  50°  to  60°  is  placed  under 
the  receiver  of  an  air  pump  and  the  air  exhausted. 
When  the  pressure  is  sufficiently  reduced,  the  water  boils 
rapidly.  (2)  Water  is  boiled  in  a  round-bottomed  flask 
until  the  air  is  expelled  by  the  steam.  The  flask  is  then 
quickly  closed  with  a  rubber  stopper  and  inverted  (Fig. 
193).  When  cold  water  is  poured  over  the  flask,  the 
water  within  it  boils  violently.  This  may  be  repeated 
till  the  water  in  the  flask  is  barely  warm.  The  cold 
water  condenses  some  of  the  vapor,  thus  decreasing  its 
pressure  upon  the  liquid. 

Owing  to  the  diminished  pressure  of  the  atmosphere 
at  high  altitudes,  the  boiling  point  of  a  liquid  is  consider- 
ably lower  upon  a  mountain  than  it  is  near  sea-level.  On 
the  summit  of  Mont  Blanc  water  boils  at  84°. 

The  following  table  gives  the  pressure  of  saturated  water 
vapor,  and  hence  also  the  pressure  under  which  water 
boils,  at  various  temperatures. 


TEMPERATURE 

PRESSURE  IN 

CM.  OF  MERCURY 

TEMPERATURE 

PRESSURE  IN 
ATMOSPHERES 

0° 

.46 

100° 

1.  00 

20 

1-74 

1  20 

1.96 

40 

5-49 

140 

3-58 

60 

14.9 

1  60 

6.12 

80 

35-5 

1  80 

9.92 

100 

76.0 

200 

15-35 

233.  Distillation.  —  A  liquid  can  be  separated  from 
non- volatile  impurities  by  boiling  it  in  a  closed  vessel,  and 
condensing  the  vapor  as  it  passes  off  through  a  tube  con- 
nected with  the  vessel.  The  process  is  called  distillation, 
and  the  apparatus  a  still.  The  vapor  is  condensed  by  in- 


276  HEAT 

closing  a  portion  of  the  tube  through  which  it  passes  within 
a  larger  tube  or  a  vessel,  in  which  it  is  surrounded  by  a 
continuous  supply  of  cold  water  (Fig.  194).  The  process 
may  be  illustrated  by  distilling  pure  water  from  a  solu- 
tion of  some  substance  whose  presence  is  shown  by  its  color, 
e.g.  copper  sulphate  or  potassium  permanganate. 


FIG.  194.  —  Distillation. 

Two  or  more  liquids  whose  boiling  points  differ  by  several  degrees 
can  be  separated  from  one  another  by  distillation.  When  such  a 
mixture  is  slowly  boiled,  the  vapor  that  passes  off  contains  a  much 
higher  percentage  of  the  more  volatile  constituent  than  the  original 
mixture  does.  Some  of  the  less  volatile  liquid  also  passes  off,  and 
complete  separation  can  be  effected  only  by  repeated  distillation. 
This  process  is  known  as  fractional  distillation.  It  is  employed  on 
a  large  scale  in  separating  the  constituents  of  crude  petroleum,  such 
as  gasoline,  naphtha,  benzin,  kerosene,  lubricating  oils,  paraffin,  etc. 

234.  Heat  of  Vaporization.  —  We  have  seen  that  heat 
energy  disappears  during  vaporization,  whether  by  evap- 
oration at  the  surface  (Art.  225)  or  within  the  liquid  (Art. 
231).  This  energy  is  recovered  as  heat  when  a  vapor 
condenses,  just  as  the  heat  of  fusion  is  recovered  when  a 


VAPORIZATION  AND   CONDENSATION  277 

liquid  freezes.  In  the  steam-heating  system,  buildings 
are  warmed  by  the  condensation  of  steam  in  the  radia- 
tors. Each  gram  of  steam,  in  condensing  to  water  at 
100°,  gives  out  more  than  five  times  as  much  heat  as  a 
gram  of  water  does  in  cooling  from  the  boiling  to  the 
freezing  point.  The  heat  generated  by  the  condensation 
of  water  vapor  in  the  atmosphere  is  the  principal  cause 
of  the  milder  temperatures  which  herald  the  approach  of 
rain  or  snow  in  winter. 

What  becomes  of  the  heat  energy  required  to  vaporize 
a  liquid?  In  the  first  place,  work  must  be  done  against 
atmospheric  pressure  in  providing  the  additional 
space  which  the  substance  occupies  as  a  vapor. 
Imagine  a  gram  of  water  to  be  placed  in  a  long 
tube",  closed  at  one  endjind  having  a  cross-section 
of  i  sq.  cm.  .Suppose- further  that  the  tube  is  fitted 
with  an  air-tight  piston,  which  moves  without  fric- 
tion (Fig.  195).  If  the  water  is  heated,  it  will  begin 
to  .vaporize  at  100°,  and  in  doing  so  will  push,  the 
piston  up.  When  vaporization  is  complete  the  piston 
will  have  been  moved  upwarcl^a  distance  of  1660  cm. 
against  the  pressure  of  the  atmosphere.  At  normal 
pressure  this  amounts  to  1033.3  g.;  and  the  steam 
must  evidently  exert  an  equal  force  against  the  FlG- 
piston.  Hence  in  making  room  for  itself  the  gram 
of  steam  does  1033.3  X  1660  =  1,715,278  g.-cm.  of  work. 
The  energy  thus  expejided  is  a  part  of  the  heat  energy 
reqinmLjo  vaporize  the  water;  and,  as  we  shall  see  later, 
it  is  the  equivalent  of  41  calories.  The  same  ^amount  of 
work  must  be  done  against  atmospheric  pressure  when  a 
gram  of  water  is  boiled  away  in  the  open  air.  The  sup- 
posed tube  and  piston  are  merely  an  aid  in  explaining  the 
process. 


278 


HEAT 


Experiment  shows,  however,  that  537  calories  are  actu- 
ally required  to  vaporize  a  gram  of  water  at  its  boiling  point. 
The  additional  496  calories  are  required  to  do  the  work  of 
separating  the  molecules  against  their  mutual  attractions, 
i.e.  in  overcoming  cohesion^  This  internal  work^is  stored 
in  Jtiie  vapor  as  molecular  potential  energy.  The  work 
done  against  atmospheric  pressure  is  called  external  work. 
Both  are  fully  recovered  as  heat  when  the  vapor  condenses. 

of  any  liquid  at  its  boiling  point  is  called  the  heat  ofjvapor- 
The  heat  of  vaporization  of  water' 
is  537  calories,  as  stated  above,  and  is  greater  than  that 
of  any  other  liquid.  For  ammonia  it  is  295  calories;  for 
alcohol,  209  calories;  for  ether,  90  calories;  and  for  mer- 
cury, 62  calories. 

235.  Heat  and  Work  in  the  Compression  and  Expan- 
sion of  Gases.  —  A  gas  does  external  work  in  expanding 
against  pressure,  just  as  a  vapor  does  in  mak- 
ing room  for  itself.  In  doing  this  work,  the 
gas  loses  an  equivalent  amount  of  heat 
energy,  and  is  cooled,  unless  it  receives  an 
equal  supply  of  heat  from  without.  This 
cooling  effect  is  shown  by  the  condensation 
of  moisture  when  saturated  air  expands  (see 
Art.  230,  under  Clouds  and  Fog).  It  can 
also  be  shown  by  inserting  the  bulb  of  a  ther- 
mometer  into  a  short  rubber  tube  through 


FIG.  196.  — Fire  which  a  jet  of  air  is  escaping  under  consider- 
able pressure  from  a  tank. 

Conversely,  a  gas  is  heated  by  compression,  the  mechan- 
ical energy  expended  upon  the  gas  being  transformed  into 
heat  within  it.  This  is  shown  in  the  heating  of  a  bicycle 


VAPORIZATION  AND   CONDENSATION  279 

pump  when  in  vigorous  use,  for  the  very  considerable 
rise  of  temperature  is  due  to  the  heat  received  from  the 
compressed  air.  When  air  is  suddenly  compressed  as  much 
as  possible  in  a  fire  syringe  (Fig.  196),  it  becomes  hot  enough 
to  ignite  a  small  piece  of  tinder,  at  the  bottom  of  the  cylin- 
der, or  vapor  of  carbon  disulphide,  mixed  with  the  air. 
The  burning  of  the  vapor  is  shown  by  a  flash  of  light. 

236.   The  Liquefaction  of  Gases.      Critical  Temperature. 

-  Within  recent  years  all  gases  have  been  liquefied,  and 
all  but  helium  reduced  to  the  solid  state.  Gases  become 
vapors  before  liquefaction  takes  place,  and  they  then  be- 
have like  other  vapors.  A  gas  can  therefore  be  liquefied 
(i)  by  cooling,  at  atmospheric  pressure,  to  a  definite  tem- 
perature, which  varies  with  different  gases,  (2)  in  some 
cases,  by  compression  at  ordinary  temperatures,  and  (3)  by 
cooling  and  compression  together.  Sulphur  dioxide  (the  gas 
formed  by  burning  sulphur)  is  liquefied  under  atmospheric 
pressure  by  a  freezing  mixture  of  ice  and  salt,  its  boiling 
point  being  -10.5°.  Under  a  pressure  of  3  atmospheres, 
it  liquefies  at  15°.  Carbon  dioxide  liquefies 'at  -80°  under 
a  pressure  of  i  atmosphere,  and  at  15°  under  a  pressure  of 
52  atmospheres. 

For  every  gas  there  is  a  certain  temperature,  called 
the  critical  temperature,  above  which  it  cannot  be  liquefied , 
however  great  the  pressure  (see  table  below).  The  further 
a  gas  is  cooled  below  its  critical  temperature  the  less  is 
the  pressure  required  for  its  liquefaction. 

In  the  manufacture  of  liquid  air,  the  air  is  compressed  to  about 
200  atmospheres,  usually  in  two  or  more  stages  to  avoid  excessive 
heating.  After  each  compression  the  air  is  cooled  by  passing  it 
through  coils  surrounded  by  water.  It  is  then  led  to  the  liquefier, 
in  which  it  flows  downward  through  the  inner  tube  of  a  double  coil. 
At  the  bottom  of  this  coil  the  air  escapes  through  a  small  opening 


280 


HEAT 


into  a  closed  space,  and  in  expanding  becomes  very  cold.  It  then 
passes  upward  through  the  outer  tube  of  the  coil,  in  which  it  surrounds 
and  cools  the  air  in  the  high-pressure  tube.  After  a  time  the  tempera- 
ture falls  so  low  that  some  of  the  compressed  air  liquefies  as  it  issues 
from  the  opening.  In  liquefying  hydrogen  the  gas  is  cooled  with 
liquid  air,  while  it  is  under  great  pressure.  The  further  cooling  due 
to  sudden  expansion  causes  it  to  liquefy.  In  1908,  Professor  Onnes,  of 
the  University  of  Leyden,  succeeded  in  liquefying  60  ccm.  of  helium 
at  5°  absolute,  70  liters  or  more  of  liquid  air  and  20  liters  of  liquid 
hydrogen  being  consumed  in  reducing  the  helium  to  this  temperature. 


SUBSTANCE 

CRITICAL 
TEMPERATURE 

BOILING  POINT 

FREEZING 
POINT 

CENT. 

ABS. 

Helium  

-242°C. 

-146 
—  140 
-119 

+    31 
130 

156 
365 

-268 
-252 

-195 
-191 
-184 
-    79 
-33-7 
-10.5 

100 

5 

21 

78 
82 
89 
194 

239-3 
262.5 

373 

-258°C. 

—  210 

-227 
-    65 

-   77 
-   76 
o 

Hydrogen  

Nitrogen  

Air 

Oxygen     

Carbon  dioxide  
Ammonia  

Sulphur  dioxide  
Water 

237.  Ice  Manufacturing  and  Cold  Storage.  —  The  rapid  evapora- 
tion of  a  highly  volatile  liquid  cools  not  only  the  liquid,  but  surround- 
ing bodies  as  well  (Arts.  225  and  226).  A  little  water  in  a  test  tube, 
placed  in  a  beaker  of  ether,  is  easily  frozen  by  evaporating  the  ether 
as  shown  in  Fig.  191.  This  principle  is  utilized  in  manufacturing 
ice  on  a  commercial  scale.  For  practical  use  the  freezing  agent  must 
be  a  liquid  whose  boiling  point  is  below  zero,  but  which  does  not 
require  an  excessive  pressure  to  liquefy  it  at  ordinary  temperatures. 
Carbon  dioxide,  ammonia  gas,  and  sulphur  dioxide  answer  these 
requirements  in  different  degree.  (See  table,  Art.  236.)  Sulphur 
dioxide  has  been  tried,  but  is  inferior  to  either  of  the  others.  Carbon 
dioxide  is  sometimes  used-,  although  a  pressure  of  800  to  900  pounds 
per  square  inch  is  required  to  liquefy  it  at  ordinary  temperatures. 
Ammonia  gas  is  generally  preferred. 


VAPORIZATION  AND   CONDENSATION 


281 


The  ammonia  sold  at  drug  stores  is  ammonia  water,  i.e.  water 
in  which  ammonia  gas  is  dissolved.  When  the  water  is  heated,  the 
gas  (NHs)  is  driven  off  in  large  volumes.  The  pure  ammonia 
exists  as  a  liquid  under  atmospheric  pressure  only  at  or  below 
~33-7°  C.  It  liquefies  at  20°  C.  when  subjected  to  a  pressure  of 
124  Ib.  per  square  inch,  or  about  8.5  atmospheres. 

Ice-making  plants  differ  from  one  another  in  more  or  less  im- 
portant details.  A  simplified  diagram  of  one  system  is  shown  in 
Fig.  197.  The  ammonia  circulates  through  a  system  of  pipes  which 
begins  and  ends  at  a  storage  tank  (not  shown  in  the  figure)  contain- 
ing a  supply  of  liquefied  ammonia  at  the  temperature  of  the  room  and 
a  pressure  of  about  10  atmospheres.  From  this  tank  the  ammonia 

Gold  Water  trickling  over  the 
ammonia  pipes  to  condense 
the  compressed  gas 


Expansion  valve 
ae  pump 

FIG.  197.  —  Artificial  Ice  and  Cold  Storage  Plant. 

flows  to  the  expansion  or  refrigerating  coils,  immersed  in  a  tank  of 
strong  brine.  A  regulating  valve  in  the  pipe  between  the  tank  and 
the  coils  permits  only  a  limited  flow;  and  the  ammonia,  on  entering 
the  coils,  rapidly  vaporizes  under  the  low  pressure  which  is  always 
maintained  in  them.  As  the  evaporation  takes  place  many  degrees 
below  zero  and  requires  a  continuous  supply  of  heat,  the  surrounding 
brine  is  also  cooled  below  zero,  and  large  cans  of  fresh  water  immersed 
in  it  are  frozen  in  from  24  to  36  hours.  The  ammonia  vapor  is  pumped 
from  the  refrigerating  coils  as  fast  as  it  forms,  and  is  driven  under  a 
pressure  of  about  10  atmospheres  through  a  set  of  condensing  coils. 
The  compression  of  the  ammonia  raises  it  to  a  high  temperature, 
and  it  must  be  cooled  before  it  will  liquefy.  This  is  accomplished 
by  allowing  water  to  trickle  over  the  condensing  coils.  From  the 
condenser  the  liquid  ammonia  is  carried  by  gravity  to  the  storage 
tank,  to  be  used  again. 


282  HEAT 

Briefly  stated,  the  purpose  of  the  ammonia  is  to  absorb  heat  at 
a  low  temperature  from  the  brine  and  to  give  it  out  at  a  high  tempera- 
ture in  the  condenser.  This  requires  an  expenditure  of  energy, 
derived  from  the  engine  which  runs  the  pump.  The  brine  used  is 
sometimes  a  strong  solution  of  table  salt,  but  as  this  is  liable  to  freeze 
from  overcooling  while  a  brine  of  calcium  chloride  is  not,  the  latter 
is  preferable. 

Artificial  cooling  or  refrigeration  is  employed  on  a  large  scale  for 
the  purpose  of  maintaining  low  temperatures  in  cold-storage  ware- 
houses, in  which  are  kept  dairy  products,  eggs,  meat,  fish,  fruits, 
etc.  Ammonia  is  used  as  the  refrigerating  agent  in  either  of  two 
ways.  In  the  brine  system  the  ammonia  cools  a  tank  of  brine,  as 
in  ice-making,  and  the  brine  is  pumped  through  coils  of  pipe,  placed 
on  the  sides  or  ceiling  of  the  room  to  be  cooled.  The  circulating 
brine  continually  absorbs  heat  from  the  cold  room  and  gives  it  to 
the  ammonia,  and  the  ammonia  carries  it  to  the  condenser.  "In  the 
direct-expansion  system  the  ammonia  is  admitted  directly  into  the 
coils  in  the  rooms  to  be  refrigerated.  The  heat  of  the  cold  room  is 
taken  up  by  the  ammonia  directly,  and  the  intermediate  agent,  brine, 
is  not  employed." 

PROBLEMS 

1.  Is  rain  water  distilled  water?     Is  it  perfectly  pure? 

2.  Water  kept  in  porous  earthenware  jars  in  warm,  dry  weather  remains 
several  degrees  below  the  temperature  of  the  air.     Explain. 

3.  At  what  depth  in  water,  exposed  to  atmospheric  pressure,  would  the 
boiling  point  be  120°? 

4.  It  takes  more  heat  to  raise  the  temperature  of  a  mass  of  gas  a  given 
number  of  degrees,  under  constant  pressure,  than  it  does  at  constant  volume. 
Why? 

5.  What  quantity  of  heat  is  required  to  convert  850  g.  of  ice  at  —  20° 
into  steam  at  ioo°? 

6.  How  much  heat  is  given  out  by  500  g.  of  steam  at  100°  in  condensing 
and  cooling  to  water  at  30°? 

7.  A  room  4  m.  by  5  m.  and  3  m.  high  is  warmed  by  a  steam  heater. 
Assuming  no  loss,  what  weight  of  steam  must  be  condensed  in  the  heater 
to  warm  the  room  from  10°  C.  to   18°  C.?     (Density  of  the  air  =  1.25  g. 
per  cu.  dm.;  specific  heat  of  air  =  .237.) 


HEATING  AND   VENTILATION  OF  BUILDINGS    283 

8.  The  nitrogen  and  oxygen  of  liquid  air  evaporate  at  unequal  rates. 
Which  evaporates  more  rapidly,  and  why? 

9.  The  temperature  of  liquid  air  does  not  rise  above  — 184°  in  an  open 
vessel.     Why  not?     How  can  it  be  made  warmer?     What  is  the  highest 
temperature  to  which  it  can  possibly  be  raised  as  a  liquid? 

10.  Why  are  icebergs  often  enveloped  in  fog? 

IX.  HEATING  AND  VENTILATION  OF  BUILDINGS 

238.  Temperature  and  Ventilating  Requirements  in 
the  Home.  —  It  is  very  generally  agreed  that  an  indoor 
temperature  of  66°  F.  during  the  winter  season  is  best 
for  persons  in  health.  Some  prefer  a  temperature  as  low 
as  64°  and  others  as  high  as  70°.  Generally  speaking, 
these  may  be  regarded  as  reasonable  limits. 

No  system  of  ventilation  will  keep  the  air  in  an  occupied 
room  as  pure  as  outdoor  air;  but  it  is  greatly  in  the 
interests  of  health  and  comfort  to  keep  it  as  pure  as 
circumstances  will  permit.  The  importance  of  this  is 
more  generally  understood  now  than  ever  before.  "We 
are  at  last  coming  to  the  conclusion  that  we  might  better 
pay  the  coal  dealer  for  the  energy  to  produce  heat, 
ventilation,  and  comfort  than  to  pay  our  physician  for 
doctoring  the  ills  resulting  from  our  carelessness." 

Competent  authorities  are  agreed  that  the  fresh-air 
supply  of  rooms  should  not  be  less  than  1800  cu.  ft.  per 
hour  for  each  occupant.  For  a  living  room  of  moderate 
size,  say  12  by  16  ft.,  this  would  require  a  complete  change 
of  air  every  hour,  with  only  one  person  in  the  room,  or  as 
many  complete  changes  per  hour  as  the  number  of  occu- 
pants. In  rooms  having  two  outside  walls,  the  leakage 
through  the  cracks  of  doors  and  windows  would  ordina- 
rily amount  to  one  or  two  changes  of  air  per  hour,  varying 
with  the  number  of  windows,  the  direction  and  velocity  of 
the  wind,  the  difference  between  inside  and  outside  tern- 


284  HEAT 

peratures,  etc.  Clearly  when  a  room  of  this  size  is  occu- 
pied by  more  than  two  persons,  the  leakage  should  not 
be  depended  upon  as  the  sole  means  of  ventilation. 

The  ventilation  of  bedrooms  is  satisfactorily  provided 
for  by  means  of  open  windows.  They  should  be  wide 
open  except  in  freezing  weather;  for  the  purer  the  air 
the  better,  and  a  low  temperature  at  night  is  in  no  wise 
harmful. 

239.  Fireplaces  and  Stoves.  —  An  open  fire  in  a  fireplace  is  a 
very  effective  and  agreeable  means  of  ventilating  a  living  room. 
The  fire  maintains  a  steady  outflow  of  air  by  way  of  the  chimney, 
and  this  quickens  the  inflow  through  the  cracks  of  doors  and  windows. 
As  a  means  of  heating  a  room  in  really  cold  weather,  a  fireplace  is 
very  inefficient  and  wasteful.  From  80  to  90%  of  the  heat  passes 
up  the  chimney;  and  the  small  remainder  is  of  little  service  except 
directly  in  front  of  the  fire,  where  the  radiation  is  most  intense.  More- 
over, the  result  is  at  best  a  compromise  between  two  uncomfortable 
extremes,  —  a  torrid  heat  on  one  side  and  an  arctic  chill  on  the  other; 
for  heating  by  radiation  is  distinctly  a  one-sided  affair.  Owing  to 
these  defects  of  the  open  fire  and  the  increasing  cost  of  fuel,  the  fire- 
place has  fallen  into  disfavor  as  the  principal  means  of  heating,  even 
in  regions  where  the  cold  of  winter  is  not  severe;  but  it  justifies  itself 
as  an  auxiliary  source  of  heat  and  a  means  of  ventilation,  aside  from 
its  attractiveness. 

The  coal  or  wood  stove  is  an  efficient  and  sanitary  means  of  heat- 
ing single  rooms,  and  offers  the  only  practical  solution  of  the  heating 
problem  for  small  country  and  town  houses.  It  diffuses  heat  both  by 
radiation  and  convection,  and  provides  a  fair  amount  of  ventilation 
by  the  action  of  the  draft,  which  withdraws  air  from  the  room  and 
carries  it  off  with  the  products  of  combustion. 

Gas  and  oil  stoves,  from  which  the  products  of  combustion  pass 
off  into  the  room,  are  highly  objectionable,  even  for  occasional  use. 
They  are  not  an  aid  to  ventilation,  and  consume  as  much  oxygen  as  a 
dozen  or  more  persons.  Where  necessity  compels  their  use,  a  window 
should  be  left  partly  open  for  ventilation,  and  the  air  should  be  com- 
pletely renewed  at  frequent  intervals  by  throwing  doors  and  windows 
wide  open. 


HEATING  AND   VENTILATION  OF  BUILDINGS    285 

240.  Furnace  Heating.  —  In  the  hot-air  system  of  heating,  the 
heater  or  furnace  is  located  in  the  basement,  and  is  inclosed  in  a  small 
brick  chamber  or  a  casing  of  sheet  iron.  A  constant  supply  of  out- 
door air  is  brought  to  the  lower  part  of  this  chamber  through  a  large 
pipe  or  duct,  and,  in  passing  over  the  hot  surfaces  of  the  furnace,  is 
heated.  It  then  rises  through  the  warm-air  pipes  at  the  top  of  the 
chamber,  and  is  discharged  through  the  registers  into  the  rooms 
above. 

In  dwellings  and  other  buildings  of  moderate  size,  the  warm-air 
registers  are  commonly  placed  in  the  floor,  or  in  the  walls  near  the 
floor,  and  cracks  in  doors  and  windows  are  depended  upon  to  provide 
the  necessary  outlet  (Fig. 
198).  The  circulation  is 
maintained  as  a  natural 
draft  or  convection  cur- 
rent, due  to  the  unequal 
densities  of  hot  and  cold 
air.  This  meets  all  re- 
quirements except  in 
windy  weather,  when 
there  is  very  likely  to  be 
trouble  in  heating  rooms 
on  the  windward  side  of 
the  house;  for  the  wind 
blows  in  through  the 


FIG.  198.  —  Hot-Air  Heating  System. 


window  crevices,  and  retards  if  it  does  not  entirely  prevent  the  inflow 
of  warm  air  from  the  registers. 

In  schoolhouses  and  other  large  buildings  the  circulation  is  main- 
tained as  a  forced  draft  by  means  of  a  centrifugal  fan  or  blower, 
placed  in  the  cold-air  duct.  A  ventilating  flue  leading  from  each 
room  provides  an  outlet  which  is  not  affected  by  the  wind.  Within 
the  room  the  air  circulates  by  convection,  the  direction  and  extent 
of  the  currents  being  determined  by  the  position  of  the  inlet  and  the 
outlet.  The  best  results  are  obtained  with  the  inlet  near  the  top 
and  the  outlet  at  the  bottom  on  the  same  side  of  the  room. 

The  hot-air  system  of  heating  is  a  great  improvement  over  the  use 
of  stoves.  It  is  comparatively  inexpensive,  and,  on  the  whole,  is 
satisfactory  in  mild  climates.  One  of  its  great  advantages  is  that  it 
insures  a  constant  supply  of  pure  air. 


286 


HEAT 


241.  Hot-water  and  Steam  Heating.  —  Hot-water  heating  is, 
on  the  whole,  most  satisfactory  for  residences  and  other  buildings  of 
moderate  size,  and  has  been  developed  to  a  high  degree  of  perfection. 
The  more  important  parts  of  the  apparatus 
are  the  heater,  located  in  the  basement, 
the  radiators  in  the  various  rooms  to  be 
warmed,  and  a  system  of  iron  pipes,  form- 
ing a  continuous  circuit  from  the  heater 
to  the  radiators  and  back  (Fig.  199).  An 
expansion  tank,  placed  above  the  highest 
radiators,  provides  for  the  expansion  and 
contraction  of  the  water.  The  pipe  lead- 
ing from  the  heater  connects  with  the  top 
of  it,  and  the  return  pipe  with  the  bottom, 
as  shown  in  the  figure.  By  this  arrange- 
ment a  gravity  circulation  (convection  cur- 
rent) is  constantly  maintained. 

Owing  to  the  great  specific  heat  of 
water,  it  is  admirably  adapted  to  serve  as 
a  medium  for  conveying  heat  from  one 
place  to  another.  A  pound  of  water  in 
cooling  through  20°  F.  (which  is  about  the 
usual  fall  of  temperature  in  the  radiator) 
gives  out  enough  heat  to  raise  the  tem- 
perature of  1 1 oo  cu.  ft.  of  air  one  degree. 
Water  is  also  well  adapted  to  meet  the 
varying  demands  of  warmer  and  colder 
weather,  as  it  can  be  heated  to  any 
desired  temperature  up  to  the  boiling 
point. 

The  hot-water  system  shown  in  the 
figure  makes  no  provision  for  ventilation. 
One  way  of  meeting  this  requirement  is 
shown  in  Fig.  200.  Outdoor  air  is  admit- 
ted through  a  duct  leading  to  the  base  of 
the  radiator,  whence  it  rises  by  convec- 
tion between  the  radiator  sections,  becoming  warm  before  it  escapes 
into  the  room. 

In  steam  heating  the  water  compartment  of  the  heater  is  only 


Fig.  199.  —  Hot- Water 
Heating  System. 


HEAT  AND  OTHER  FORMS  OF  ENERGY 


287 


FIG.  200.  —  Heating 
with  Ventilation. 


partly  filled,  and  the  water  is  boiled.  The  steam  forces  its  way 
through  the  pipes  to  the  radiators,  where  it  gives  out  heat  and  con- 
denses, returning  as  hot  water  to  the  boiler.  A  safety  valve  at 
the  boiler  guards  against  excessive  pressure,  taking  the  place  of 
the  expansion  tank  in  hot-water  heating.  The 
same  pipe  can  be  made  to  serve  both  for  the 
flow  of  the  steam  and  the  return  of  the  hot 
water,  the  steam  taking  the  upper  side  of  the 
pipe  and  the  water  the  lower.  The  weight  of 
steam  required  to  deliver  a  given  amount 
of  heat  to  the  radiator  is  only  about  one 
fiftieth  as  great  as  the  weight  of  water  re- 
quired to  accomplish  the  same  result  by  the 
hot-water  system;  for  each  gram  of  steam 
in  condensing  gives  out  537  calories,  while  a 
gram  of  hot  water  in  cooling  through  20°  F. 
gives  out  only  n  calories.  Steam  radia- 
tors are  about  two  thirds  as  large  as  hot- 
water  radiators  of  the  same  heating  capacity;  for  steam  keeps 
the  radiating  surfaces  at  212°  F.,  while  with  water  the  average  is 
about  170°  F. 

The  relative  merits  of  steam  and  hot-water  heating  depend  largely 
upon  the  requirements  to  be  met  in  any  given  case.  In  general,  hot- 
water  heating  is  preferable  for  residences  and  other  buildings  of 
moderate  size,  and  steam  heating  for  large  apartment  houses,  busi- 
ness buildings,  churches,  public  halls,  etc. 

X.  HEAT  AND  OTHER  FORMS  OF  ENERGY 

242.  The  Mechanical  Equivalent  of  Heat.  —  If  energy 
is  never  created  or  destroyed  (Art.  153),  a  definite  numer- 
ical relation  must  exist  between  any  given  amount  of  energy 
of  one  kind  and  the  equivalent  amount  of  energy  of  an- 
other kind.  Thus  a  certain  number  of  gram-centimeters  of 
mechanical  energy  must  always  be  required  to  generate 
one  calorie  of  heat,  in  whatever  way  the  transformation 
may  be  brought  about.  This  number  is  called  the  mechan- 
ical equivalent  of  heat.  It  is  of  very  great  scientific  and 


288 


HEAT 


practical  importance,  and  has  been  carefully  determined 
in  various  ways  by  different  physicists.  James  Prescott 
Joule,  of  Manchester,  England,  was  the  first  to  establish 
the  fact  that  such  a  relation  exists  and  to  determine  its 
value.  His  numerous  experiments  extended  over  several 
years  (1843  to  ^So),  and  embraced  several  different 
methods.  In  each  case  a  measured  quantity  of  mechanical 
or  electrical  energy  was  converted  into  a  measured  quan- 
tity of  heat.  These  notable  experiments  settled  the  long 
dispute  concerning  the  nature  of  heat  (Art.  184),  and  af- 
forded a  sure  foundation  for  the  doctrine  of  the'  conserva- 
tion of  energy. 

The  main  features  of  the  method  which  Joule  preferred  are  as 
follows:  By  an  arrangement  of  wheels  and  axles  shown  in  Fig.  201, 

two  heavy  weights,  e  and  e,  turn 
a  set  of  paddles  in  a  calorimeter 
filled  with  water.  Stationary 
projections,  extending  inward 
between  the  paddles  from  the 
sides  of  the  calorimeter,  prevent 
the  water  from  revolving  bodily 
with  the  paddles  and  keep  it  in 
violent  agitation.  The  work 
done  by  the  weights  in  falling 
is  converted  into  heat  by  the  internal  friction  of  the  water.  This 
work  is  measured  in  gram-centimeters  by  the  product  of  the  force 
of  gravity  upon  the  weights  and  the  distance  through  which  they 
descend.  The  number  of  calories  generated  is  computed  from  the 
weight  of  the  water  and  the  calorimeter,  the  specific  heat  of  the 
calorimeter,  and  the  rise  of  temperature.  Hence,  after  making 
necessary  allowances  for  conduction,  radiation,  etc.,  the  equivalent 
of  a  certain  number  of  calories  is  obtained  in  gram-centimeters  of 
mechanical  energy;  from  which  the  equivalent  of  one  calorie  is 
computed. 

The  value  now  accepted  for  this  equivalent,  after  re- 
peated determinations  by  different  physicists,  is  42,700 


FIG.  201.  —  Joule's  Apparatus. 


HEAT  AND   OTHER  FORMS  OF  ENERGY         289 

g.-cm.;  that  is,  42,700  g.-cm  of  mechanical  energy  will 
generate  one  calorie  when  wholly  transformed  into  heat, 
and  vice  versa. 

243.  Sources  of  Heat.  —  All  other  forms  of  energy  can 
become  heat,  directly  or  indirectly,  by  various  transforma- 
tions,  many  of  which   have   already   received   attention 
(Arts.  151,  152,  184).     There  is,  as  we  have  seen  (Art. 
154),  a  natural  tendency  for  such  transformations  to  take 
place ;  but  in  most  cases  the  heat  generated  is  of  no  impor- 
tance as  heat  and  is  not  available  for  doing  useful  work. 
The  only  useful  sources  of  heat,  on  a  large  scale,  are  (i) 
solar  radiation,  (2)  the  chemical  energy  of  fuels  and  foods, 
and  (3)  electrical  energy. 

Electrical  energy  as  a  source  of  heat  will  be  considered 
under  the  general  subject  of  electricity.  The  heat  ob- 
tained from  fuels  is  generated  during  chemical  change, 
in  which  the  molecules  of  the  burning  substance  break  up, 
and  their  constituent  parts  (atoms)  unite  with  oxygen  from 
the  air.  New  substances  are  thus  formed,  principally 
gases,  whose  molecules  are  generally  less  complex  than  those 
of  the  fuel  and  always  possess  less  energy. 

244.  Solar  Energy  and  its  Transformations.  —  The  sun  is  the 
original  source  of  practically  all  available  energy,  and  this  energy  is 
directly  or  indirectly  the  cause  of  nearly  all  terrestrial  phenomena. 
The  only  exceptions  of  any  magnitude  are  the  tides  and  phenom- 
ena due  to  the  slow  shrinking  of  the  earth  and  to  the  heat  of  its 
interior,  such  as  volcanic   action,   earthquakes,    the   formation  of 
mountains,  etc.     Although  the  interior  of  the  earth  is  intensely  hot, 
the  heat  is  conducted  to  the  surface  so  slowly  that  its  effect  upon 
the  temperature  of  the  atmosphere  is  inappreciable. 

The  energy  of  winds  is  directly  traceable  to  the  sun,  for  the  winds 
are  due  to  the  unequal  heating  of  different  portions  of  the  earth's 
surface.  Water-power  has  the  same  ultimate  source;  for  it  is  by 
means  of  solar  radiation  that  water  is  evaporated  from  the  oceans 
and  carried,  through  the  agency  of  winds,  to  the  highest  mountains. 


2  go  HEAT 

Plants  take  the  materials  necessary  for  their  growth  from  the 
earth  and  air;  but  their  energy  is  received  directly  from  the  sun. 
The  leaves  absorb  carbon  dioxide  from  the  air,  and,  under  the  action 
of  sunlight,  separate  it  into  its  constituents  (carbon  and  oxygen). 
The  carbon  unites  with  water  sent  up  from  the  roots,  forming  starch; 
the  oxygen  is  given  off  to  the  air  again.  From  the  starch  and  various 
earthy  materials  absorbed  with  water  from  the  soil,  the  more  complex 
substances  are  formed  which  are  needed  for  the  growth  of  the  plant. 
These  substances  possess  energy  which  comes  from  sunlight  absorbed 
by  the  green  coloring  matter  in  the  leaves  (the  chlorophyll).  Radi- 
ant energy  is  thus  transformed  and  stored  as  chemical  potential  energy 
in  the  substance  of  the  plant  itself.  The  energy  of  coal  is  also  stored 
solar  energy;  for  the  coal-beds  are  the  remains  of  immense  forests 
that  grew  long  before  man  appeared  upon  the  earth. 

Animals,  as  already  noted  (Art.  152),  derive  the  energy  for  all 
their  bodily  activities  from  their  food,  and  hence  originally  from  the 
\  sun,  whether  the  food  be  of  vegetable  or  of  animal  origin. 

245.  Amount  of  Solar  Radiation.  —  From  a  law  of  radiation  it  is 
known  that  the  amount  of  energy  radiated  from  any  portion  of  the 
sun's  surface  is  46,000  times  as  great  as  that  received  by  an  equal 
area  of  the  earth's  surface  in  the  same  time.     The  energy  received 
has  been  approximately  measured;  hence  the  rate  at  which  the  sun  is 
giving  out  energy  is  known  to  the  same  approximation.     This  amounts 
to  nearly  100,000  horse-power  per  square  meter  of  the  sun's  surface, 
acting  continuously.     To  maintain  this  rate  of  radiation  by  combus- 
tion "  would  require  the  hourly  burning  of  a  layer  of  the  best  anthra- 
cite coal  from  sixteen  to  twenty  feet  thick  over  the  sun's  entire 
surface,  —  a  ton  for  every  square  foot  of  surface,  —  at  least  nine  times 
as  much  as  the  most  powerful  blast  furnace  in  existence.     At  that 
rate  the  sun,  if  made  of  solid  coal,  would  not  last  6000  years."  *     Of 
this  enormous  output  of  energy  the  earth  receives  only  one  part  in 
twenty-two  hundred  million. 

246.  Source  of  the   Sun's  Energy.  —  The  source  of  the  sun's 
energy  is  a  question  of  the  greatest  scientific  interest.     A  direct 
answer  not  being  obtainable,  various  theories  have  been  suggested, 
all  of  which  recognize  the  principle  of  the  conservation  of  energy. 

*  Young's  General  Astronomy. 


HEAT  AND  OTHER  FORMS  OF  ENERGY    291 

The  sun's  heat  can  not  be  maintained  by  combustion,  for  in  that 
case  it  would  have  been  burned  out  long  ago.  "  Nor  can  it  be  simply 
a  heated  body  cooling  down.  Huge  as  it  is,  an  easy  calculation  shows 
that  its  temperature  must  have  fallen  greatly  within  the  last  2000  years 
by  such  a  loss  of  heat,  even  if  it  had  a  specific  heat  higher  than  that 
of  any  known  substance."  * 

The  theory  generally  accepted  is  known  as  Helmholtz's  theory  of 
solar  contraction.  This  is  that  "  the  heat  necessary  to  maintain 
the  sun's  radiation  is  principally  supplied  by  the  slow  contraction 
of  its  bulk,  aided,  however,  by  the  accompanying  liquefaction  and 
solidification  of  portions  of  its  gaseous  mass.  When  a  body  falls 
through  a  certain  distance  gradually,  against  resistance,  and  then 
comes  to  rest,  the  same  total  amount  of  heat  is  produced  as  if  it  had 
fallen  freely,  and  been  stopped  instantly.  If,  then,  the  sun  does 
contract,  heat  is  necessarily  produced  by  the  process,  and  that  in 
enormous  quantity,  since  the  attracting  force  at  the  solar  surface  is 
more  than  twenty-seven  times  as  great  as  terrestrial  gravity,  and 
the  contracting  mass  is  immense.  Helmholtz  has  shown  that,  under 
the  most  unfavorable  conditions,  a  contraction  in  the  sun's  diameter 
of  about  two  hundred  and  fifty  feet  a  year  would  account  for  the 
whole  annual  output  of  heat."  *  At  this  rate,  a  period  of  9000  years 
would  be  required  for  a  total  contraction  sufficiently  great  to  be 

detected  by  measurement  with  the  best  astronomical  instruments. 

• 

PROBLEMS 

1.  Compute  the  rise  of  temperature  that  would  be  obtained  with  Joule's 
apparatus  under  the  following  conditions:  Mass  of  the  weights  used  =  30  kg. 
Distance  through  which  the  weights  descend  =  25  m.     Weight  of  water  in 
the  calorimeter  =  2  kg.     Weight  of  the  copper  calorimeter   =  i  kg. 

2.  (a)  A  mass  of  iron  weighing  i   kg.  falls  upon  a  stone  from  a  height 
of  100  m.     How  much  heat  is  generated,  assuming  that  the  energy  is  all 
transformed  into  heat?     (b)  If  half  of  the  heat  is  generated  in  the  mass 
of  iron,  what  is  its  rise  of  temperature? 

3.  From  what  height  must  a  mass  of  iron  fall  in  order  that  the  heat  gen- 
erated when  it  strikes  shall  be  sufficient  to  raise  its  temperature  one  degree, 
assuming  that  the  energy  is  all  transformed  into  heat  in  the  iron  itself? 

4.  A  lead  bullet  strikes  a  target  with  a  velocity  of  300  m.  per  second. 
Assuming  that  90%  of  its  energy  is  transformed  into  heat  in  itself,  what  is 
its  rise  of  temperature? 

*  Young's  General  Astronomy. 


292  HEAT 

XI.  HEAT  ENGINES 

247.  Fundamental  Principle.  —  The  transformation  of 
heat  into  mechanical  energy  takes  place  on  a  large  scale 
only  in  the  vaporization  of  liquids  (Art.  234)  and  in  the 
expansion  of  gases  and  vapors  (Art.  235).     Any  machine 
or  device  by  means  of  which  this  transformation  is  con- 
trolled and  made  to  do  useful  work  is  called  a  heat  engine. 
In  this  general  sense  a  cannon  is  a  heat  engine.     When  a 
cannon  is  fired,  the  chemical  energy  of  the  powder  becomes 
heat  energy  in  the  gases  generated  by  the  explosion.     At 
the  instant  of  the  explosion  the  gases  are  intensely  hot; 
but,  in  expanding  under  great  pressure,  their  heat  is  largely 
transformed  into  the  kinetic  energy  of  the  projectile,  and 
they  issue    from   the  mouth  of  the  cannon    reduced  in 
temperature. 

Mechanical  energy  can  be  wholly  converted  into  heat; 
but  the  reverse  transformation  is  always  partial.  The 
best  that  any  heat  engine  can  do  is  to  utilize  the  heat  given 
out  by  steam  or  gases  in  expanding  and  cooling  through  a 
greater  or  less  range  of  temperature. 

248.  The  Steam  Engine  converts  heat  into  mechanical 
energy  through  the  expansive  power  of  steam  generated 
under  pressure.     A  simplified  diagram  of  its  working  parts 
is  shown  in  Fig.  202.    While  the  engine  is  running,  the  steam 
chest  receives  a  constant  supply  of  steam  through  a  pipe 
leading  from  a  boiler.     This  steam  is  admitted  into  the 
cylinder  through  narrow  passages,  called  ports,  which  are 
alternately  opened  and  closed  by  a  slide-valve.     In  the 
position  shown  in  the  figure,  the  steam  enters  at  the  upper 
end  of  the  cylinder,  and  pushes  the  piston  and  the  piston- 
rod  down.     Meanwhile  the  slide-valve  rises,  shutting  off 
the  steam  from  the  upper  end  of  the  cylinder,  while  the 


HEAT  ENGINES 


293 


piston  is  still  moving  downward,  and  admitting  it  into  the 
lower  end  just  as  the  stroke  is  completed.  The  piston  is 
now  driven  upward,  while  the  slide-valve  descends.  At 
the  end  of  this  stroke  the  valve  and  the  piston  are  again  in 
the  positions  shown  in  the  figure.  As  the  valve  moves  to 
admit  steam  at  either  end  of  the  cylinder,  it  connects  the 
port  at  the  other  end  with  an  exhaust  pipe,  through  which 
the  spent  steam  on  that  side  of  the 
piston  escapes,  under  reduced  pressure, 
into  the  open  air  or  into  a  condens- 
ing chamber  (Art.  250). 

The  to-and-fro  motion  of  the  piston 
rod  causes  the  connecting  rod  to  turn 
a  shaft.  The  shaft  carries  a  flywheel 
to  maintain  uniform  rotation  (Art. 
150),  and  is  connected,  generally  by 
means  of  a  belt,  with  the  machinery 
to  be  operated  by  the  engine.  The 
shaft  also  carries  an  eccentric,  which 
imparts  the  necessary  to-and-fro  mo- 
tion to  the  valve-rod. 


FIG.  202.  —  Section  of 
Steam  Engine. 


249.  Efficiency  Gained  by  an  Early 
Cut-off.  —  If  the  steam  were  admitted 
to  the  cylinder  of  an  engine  during 
the  entire  stroke  of  the  piston,  only  a  fraction  of  its  avail- 
able energy  would  be  used;  for  it  would  still  be  capable 
of  expanding  and  doing  work.  The  greatest  efficiency 
under  ordinary  conditions  is  secured  by  adjusting  the  slide- 
valve  so  as  to  cut  off  the  supply  of  steam  at  or  near  the 
first  quarter  of  the  stroke.  The  steam  then  in  the  cylinder 
works  expansively  to  the  end  of  the  stroke;  and,  in  expand- 
ing, its  temperature  and  pressure  fall  rapidly.  For  example, 


294 


HEAT 


if  the  steam  is  admitted  under  a  pressure  of  6  atmospheres 
and  discharged  into  the  air,  its  temperature  falls  from  about 
160°  C.  to  100°  (see  Table,  Art.  232).  The  heat  lost  by 
the  steam  in  expanding  is  converted  into  mechanical  en- 
ergy in  driving  the  piston.  (There  is,  of  course,  some  loss 
of  heat  to  the  walls  of  the  cylinder.) 

250.  Condensing  Engines.  —  The  effective  or  resultant 
pressure  against  the  piston  of  an  engine  is  the  difference 
between  the  pressure  of  the  new  steam  on  the  one  side  and 
the  back  pressure  of  the  exhaust  steam  on  the  other.  When 
the  steam  is  exhausted  into  the  air,  this  back  pressure 
is  necessarily  somewhat  greater  than  that  of  the  air,  i.e. 
15  Ib.  or  more  per  square  inch.  Thus  if  the  average  pres- 
sure of  the  working  steam  during  a  complete  stroke  is  60 
Ib.  (absolute)  per  square  inch,  the  average  effective  pressure 
against  the  piston  is  less  than  45  Ib.  per  square  inch,  and 
one  fourth  of  the  energy  that  would  be  available  if  the 
steam  were  exhausted  into  a  vacuum  is  lost  in  working 
against  atmospheric  pressure. 

To  avoid  this  waste,  the  exhaust  pipe 
of  stationary  and  marine  engines  usually 
leads   to   a    condensing    chamber    (Fig. 
203),    in    which    a    partial    vacuum    is 
maintained  by  means  of  a   pump.     As 
the  exhaust  steam  enters  this   chamber 
it  is  quickly  condensed  by  a   spray  of 
cold  water,  as  shown  in  the  figure,  or 
by    water    surrounding     the     chamber, 
ing  Chamber.        The   vacuum    mamtained    in    the    con- 
denser reduces  the  back  pressure  of  the  exhaust  steam  by 
approximately  one  atmosphere,  and  hence  increases  the 
effective  pressure  against  the  piston  by  that  amount. 


FIG.  203.  —  Condens- 


HEAT  ENGINES 


295 


251.  Compound  Engines.  —  The  available  energy  of  steam  in- 
creases with  its  pressure.  At  a  gage  pressure  of  10  Ib.  (i.e.  10  Ib. 
per  square  inch  above  atmospheric  pressure),  a  perfect  non-condensing 
engine  uses  69  Ib.  of  steam  per  horse-power  per  hour.  It  does  an 
equal  amount  of  work  with  26  Ib.  of  steam  at  40  Ib.  pressure,  with  18  Ib. 
at  80  Ib.  pressure,  with  16.3  Ib.  at  100  Ib.  pressure,  or  with  13.6  Ib. 
at  150  Ib.  pressure. 

The  additional  amount  of  heat  required  to  generate  steam  at  high 
pressures  is  relatively  small,  compared  with  the  gain  of  available 
energy.  Hence  there  is  economy  of  fuel  as  well  as  of  steam  in  work- 
ing the  steam  at  high  pressure.  But  the  highest  efficiency  of  steam 
at  any  pressure  is  secured  only  when  the  full  expansive  power  of  the 
steam  is  utilized;  and  the  greater  the  initial  pressure  the  greater  the 
possible  expansion.  When  the  pressure  is  above  100  Ib.,  the  greatest 
efficiency  is  obtained  by  using  the  steam  successively  in  two  or  more 
cylinders  (Fig.  204),  with  not  more  than  a  threefold  or  fourfold 


FIG.  204.  —  Section  of  a  Duplex  Compound  Engine. 

expansion  in  each.  The  exhaust  from  the  first  or  high-pressure 
cylinder  passes  into  the  larger  low-pressure  cylinder,  in  which  it 
drives  a  second  piston.  Such  engines  are  called  compound.  Large 
marine  and  stationary  engines,  working  under  the  highest  pressures 
(200  to  225  Ib.),  use  the  steam  successively  in  three  or  four  cylinders. 
These  are  called  triple-expansion  and  quadruple-expansion  engines 
respectively. 

The  cylinders  of  a  compound  engine  are  sometimes  placed  side 
by  side,  as  in  the  accompanying  illustration,  sometimes  end  to  end. 


296  HEAT 

The  latter  arrangement  is  common  with  two-cylinder  engines,  and 
is  known  as  the  tandem  compound.  In  such. engines  the  two  piston 
heads  are  attached  to  the  same  rod.  A  twin-tandem  compound 
consists  of  two  tandem  compounds,  built  as  one  unit  for  driving  the 
same  shaft.  Compound  engines  are  usually  condensing  engines. 

The  various  types  of  compound  condensing  engines  represent  the 
highest  development  of  the  modern  steam-engine.  They  furnish 
the  power  for  propelling  steamships  and  for  the  heaviest  duty  in 
manufacturing  establishments.  Engines  ranging  in  power  from 
1000  to  5000  horse-power  are  common.  The  largest  steam-engine 
in  the  world  is  installed  in  a  rolling-mill  at  Sharon,  Pennsylvania. 
It  is  a  twin-tandem  engine,  having  high-pressure  cylinders  42  in.  in 
diameter  and  low-pressure  cylinders  70  in.  in  diameter.  It  weighs 
550  tons  without  foundation  plates  or  flywheel,  and  is  rated  at 
25,000  horse-power. 

252.  The  Locomotive.  —  The  main  body  of  a  locomotive  consists 
of  the  long,  horizontal  boiler,  the  cab  back  of  it,  the  fire  box  under  its 

rear  end,  and  the  smoke 
box  in  front  of  it.  The 
boiler  contains  from  300 
to  400  tubes,  extending 
from  the  fire  box  to  its 
front  end  (Fig.  205). 
These  convey  the  hot 

gases  from  the  fire  to 
FIG.  205.  — Tubular  Boiler.  , 

the  smoke  box,  and  pro- 
vide a  large  heating  surface,  being  entirely  surrounded  by  the  water  in 
the  boiler.  A  locomotive  has  at  least  two  cylinders,  one  on  each  side, 
below  the  level  of  the  boiler  at  the  forward  end.  The  piston  working 
in  each  cylinder  acts  on  the  driving  wheels  on  its  own  side.  The 
cylinders  are  supplied  with  steam  through  a  pipe  which,  starting 
within  the  steam  dome,  runs  forward  through  the  boiler  and  down 
through  the  smoke  box.  The  throttle  valve  at  the  entrance  to  this 
pipe  serves  to  regulate  the  pressure  of  the  steam  admitted  to  the 
cylinders.  The  ports  of  each  cylinder  are  opened  and  closed  by 
means  of  a  slide  valve,  as  in  stationary  engines.  The  adjustment 
of  this  valve  is  under  the  immediate  control  of  the  engineer,  through 
a  system  of  eccentrics  and  levers,  ending  with  the  reversing  lever  in 


HEAT  ENGINES 


297 


the  cab.  When  this  lever  is  in  a  forward  position  the  engine  runs 
forward;  when  it  is  inclined  to  the  rear  the  engine  runs  backward; 
when  it  is  in  a  vertical  position  the  valve  remains  at  rest  with  both 
ports  closed,  and  the  engine  coasts,  without  working.  The  point  of 
cut-off  of  the  steam  is  determined  by  the  greater  or  less  inclination 
of  the  lever.  For  most  economical  working  the  cut-off  occurs  at 
quarter  stroke;  for  greater  power  it  occurs  at  half  stroke. 

Locomotives  are  non-condensing  engines.  The  exhaust  takes 
place  into  the  smoke  box,  just  below  the  smokestack.  This  maintains 
a  powerful  draft  for  the  fire,  and  is  the  cause  of  the  familiar  puffing 
sound.  Express  locomotives,  being  built  for  speed,  have  large  driving 
wheels  from  6  to  7  ft.  in  diameter;  freight  locomotives  are  designed 
for  great  tractive  power,  their  drivers  having  a  diameter  of  5  or  6 
ft.  The  most  powerful  locomotives  are  compound  and  have  from 


FIG.  206.  —  Diagram  of  a  Locomotive  Engine. 

8  to  1 6  driving  wheels.  The  largest  are  capable  of  exerting  a  tractive 
effort  or  pull  at  the  draw-bar,  of  80,000  to  100,000  Ib.  The  trac- 
tive effort  of  the  average  express  locomotive  is  from  16,000  to  25,000 
Ib.;  of  the  average  freight  locomotive,  from  30,000  to  45,000  Ib. 
The  tractive  effort  averages  about  one  fifth  of  the  weight  carried  by 
the  drivers.  If  it  is  greater  than  one  fourth,  the  wheels  are  liable  to 
slip.  The  usual  boiler  pressure  of  locomotives  is  about  200  Ib.,  the 
maximum  225  Ib.  . 

253.    Efficiency   of    the    Steam-Engine.  —  The   various   modern 
types  of  the  steam-engine  are  highly  perfected  machines;  and  yet 


298  HEAT 

the  entire  process  of  converting  the  energy  of  fuel  into  useful  work 
through  the  agency  of  steam  is  a  very  wasteful  one,  only  1 2%  of  the 
energy  being  thus  converted  under  the  most  favorable  conditions.  In 
the  first  place,  "  from  one  fourth  to  one  fifth  of  the  heat  produced  by 
combustion  in  the  furnace  is  lost  or  carried  up  the  chimney  in  the 
gases  at  the  high  temperature,  besides  the  waste  due  to  radiation, 
and  smoke  imperfectly  burned."  But  this  is  not  the  greatest  waste. 
At  least  three  fourths  of  the  energy  of  high-pressure  steam  is  carried 
away  in  the  exhaust,  when  this  loss  is  reduced  as  much  as  possible  by 
means  of  a  condenser.  This  is  the  heat  of  vaporization,  and  it  can 
not  be  utilized  in  any  engine. 

The  additional  energy  put  into  steam  by  generating  it  under  great 
pressure  and  by  superheating  it  is  all  available  for  doing  work,  and 
adds  greatly  to  the  efficiency  of  an  engine.  Steam  is  superheated, 
after  leaving  the  boiler,  by  passing  it  through  pipes  which  are  sur- 
rounded by  the  hot  gases  from  the  furnace.  .  Ordinary  or  saturated 
steam  begins  to  condense  as  soon  as  it  begins  to  expand  in  the  cylinder 
of  an  engine;  superheated  or  unsaturated  steam  does  not  (Art.  222). 
The  thermal  efficiency  of  an  engine  is  the  ratio  of  the  work  done  upon 
the  piston  to  the  energy  supplied  in  the  steam.  The  best  compound 
condensing  engines,  when  using  highly  superheated  steam,  have  a 
maximum  thermal  efficiency  of  about  17%. 

Lastly,  from  one  tenth  to  one  fifth  of  the  work  done  on  the  piston 
is  lost  or  absorbed  by  friction  of  the  engine  mechanism,  and  only 
80  to  90%  of  it  is  transmitted  to  the  driving  shaft.  The  percentage 
of  work  transmitted  is  called  the  mechanical  efficiency  of  the  engine. 
The  highest  economic  efficiency  of  an  entire  steam  plant,  including 
furnace,  boiler,  and  engine,  is  therefore  about  .8  X  .17  X  .9  =  12%, 
and  6%  is  a  fair  average  value.  The  efficiency  of  locomotives  is 
considerably  less,  rarely  if  ever  exceeding  6%. 

254.  Historical  Notes  on  the  Steam-Engine.  —  The  earliest 
steam-engine  having  a  cylinder  and  piston  was  invented  in  1705  by 
Thomas  Newcomen,  an  English  blacksmith.  The  cylinder  was 
vertical,  and  its  upper  end  open.  The  piston  was  driven  up,  against 
atmospheric  pressure,  by  steam  admitted  into  the  lower  end  of  the 
cylinder.  The  steam  in  the  cylinder  was  then  condensed  by  a  jet 
of  water.  A  partial  vacuum  was  thus  formed  under  the  piston, 
which  was  then  driven  down  by  atmospheric  pressure.  The  supply 


HEAT  ENGINES 


299 


of  steam  and  water  was  controlled  by  opening  and  closing  stop  cocks, 
which  were  at  first  operated  by  hand.  Newcomen  engines  were 
successfully  used  for  pumping  water  from  mines. 

Such  was  the  steam-engine  when  James  Watt,  instrument  maker 
to  the  University  of  Glasgow,  began  its  improvement  in  1768.  He 
introduced  a  separate  condenser,  admitted  steam  to  both  ends  of 
the  cylinder,  and,  during  a  period  of  thirty  years  or  more,  made  many 
other  improvements  which  greatly  increased  its  efficiency.  Many 
inventors  have  taken  part  in  the  development  of  the  steam-engine 
since  the  time  of  James  Watt;  but  to  him,  more  than  to  any  other 
man,  is  due  the  honor  of  having  made  it  one  of  the  great  factors  in 
the  industrial  progress  of  the  world. 

The  first  self-moving  steam-engine  was  built  in  France  in  1769; 
the  first  in  America  was  built  in  1790.  Both  were  designed  to  run 
on  common  roads.  Railroad 
locomotives  were  first  built 
and  successfully  operated  in 
England.  The  first  "  prac- 
tical "  American  locomotive 
was  the  "Tom  Thumb," 
built  by  Peter  Cooper,  of 
New  York,  in  1831.  "That 
engine  developed  about  one 
and  one  half  horse-power, 
and  the  chief  objection  raised 

*— ^^5^ 


FIG.  207. — Locomotive  of  1839. 


was  that  it  was  not  powerful 
enough."  That  has  been  a 
common  objection  to  the 
most  powerful  engines  ever  since. 

Navigation  by  steam  power  began  to  be  a  success  in  1807,  when 
the  little  steamboat  Clermont,  constructed  under  the  direction  of 
Robert  Fulton,  made  its  first  trip  from  New  York  to  Albany. 

255.  The  Internal  Combustion  Engine.  —  The  problem 
of  devising  an  engine  that  could  be  made  to  do  useful 
work  by  the  combustion  of  fuel  within  its  cylinder  was 
attempted  even  before  the  invention  of  the  steam-engine. 
It  was  first  proposed  to  drive  the  piston  by  the  explosion 


300 


HEAT 


of  gunpowder;  but  the  first  practical  success  was  achieved 
in  1860,  with  coal  gas  as  fuel.  Since  then  improvements 
have  followed  in  rapid  succession,  the  most  important 
being  the  adoption  of  the  four-stroke  cycle  by  the  German 
inventor,  Dr.  Otto,  in  1876.  Modern  gas  and  gasoline 
engines  are  principally  of  the  Otto  type,  and  are  known 
as  four-cycle,  or,  more  correctly,  four-stroke-cycle  engines 
(Fig.  208). 

The  Otto  cycle  consists  of  a  series  of  operations  which 
take  place  during  four  successive  strokes  of  the  piston, 

or  two  complete  revolutions 
of  the  flywheel.  These  opera- 
tions are  as  follows:  First  or 
suction  stroke;  charging.  The 
piston  moves  forward  or  out- 
ward (downward  in  vertical 
engines),  drawing  in  a  charge 
of  air  and  gas  through  the 
inlet  valve  /.  The  exhaust 
valve  remains  closed.  Second 
stroke;  compression.  During 
the  return  or  inward  stroke, 
both  valves  are  closed,  and 
c,  Working  stroke;  D,  Exhaust;  the  charge  admitted  during 

/,  Inlet  Valve;  E,  Exhaust  Valve.     -,...-  ,       . 

the  first  stroke  is  compressed. 
Third  or  working  stroke;  explosion  and  expansion.  The 
compressed  mixture  is  ignited  by  an  electric  spark  or 
other  device.  It  explodes,  forming  other  gases  at  a  very 
high  temperature  and  a  correspondingly  high  pressure. 
These  gases  expand,  driving  the  piston  outward,  and  the 
work  done  on  the  piston  is  transmitted  through  the  con- 
necting rod  to  the  crank-shaft.  This  is  the  only  part  of 
the  cycle  in  which  heat  is  converted  into  work;  hence  it 


1 


HEAT  ENGINES  301 

is  called  the  working  or  power  stroke.  Fourth  stroke;  ex- 
haust. During  the  second  inward  stroke  the  piston  drives 
out  the  products  of  combustion  through  the  exhaust  valve 
E.  This  completes  the  cycle. 

The  piston  of  an  internal  combustion  engine  is  generally  driven 
from  one  side  only,  the  forward  end  of  the  cylinder  being  open  to  the 
air.  Since  there  is  only  one  working  stroke  in  four,  the  flywheel 
must  be  very  massive  in  order  to  maintain  steady  motion.  There 
are  usually  two  flywheels,  one  on  each  side.  Small  engines  are 
started  by  hand,  large  ones  usually  by  compressed  air.  The  valves 
are  circular  and  fit  circular  openings,  against  which  they  are  tightly 
held  by  spiral  springs.  They  open  with  an  inward  motion,  and  are 
operated  from  the  crank-shaft  by  a  rather  complicated  mechanism 
of  gear-wheels,  rods,  cams,  and  levers.  Since  the  cylinder  is  directly 
exposed  to  the  intense  heat  of  the  burning  gases,  some  effective  means 
must  be  provided  for  cooling  it.  The  cylinders  of  very  small  engines 
are  cast  with  numerous  projecting  flanges  or  ribs,  which  present  a 
large  cooling  surface  to  the  air.  In  the  larger  sizes  the  cylinder  is 
surrounded  by  a  jacket,  through  which  water  is  kept  in  constant 
circulation. 

256.  Types  of  Internal  Combustion  Engines.  —  There  are  many 
forms  of  the  internal  combustion  engine,  varying  with  the  kind  of 
fuel  used,  the  power  of  the  engine,  and  the  purpose  for  which  it  is 
intended. 

Stationary  Gas  Engines  for  general  power  purposes  are  both  ver- 
tical and  horizontal.  Those  of  moderate  power  have  one  cylinder; 
the  largest  are  tandem  or  twin-tandem  engines,  having  two  and  four 
cylinders  respectively,  and  ranging  from  500  to  5000  horse-power. 
The  smaller  engines  are  run  with  illuminating  gas,  the  larger  with 
a  cheaper  gas,  generated  in  "  gas  producers  "  from  crude  oil,  inferior 
grades  of  coal,  or  other  inexpensive  fuels.  The  gas  producer  is  to 
the  gas  engine  what  the  boiler  is  to  the  steam-engine. 

A  power  plant,  consisting  of  a  modern  gas  engine  and  gas  producer, 
converts  from  20  to  25%  of  the  heat  energy  of  the  fuel  into  useful 
work,  the  average  efficiency  being  two  or  three  times  as  great  as  that 
of  a  steam-engine  plant  of  the  same  capacity. 

The  Marine  Gas  Engine.  —  The  use  of  the  gas  engine  for  propelling 


302 


HEAT 


ships  is  still  in  the  experimental  stage,  but  success  in  the  near  future 
is  very  probable.  The  principal  advantages  offered  by  the  marine 
gas  engine  are  a  higher  efficiency,  a  considerable  saving  of  space  and 
weight,  and  the  fact  that  smokestacks  would  be  done  away  with. 

The  Gasoline  Engine     (Fig.  209) 
is  driven  by  an  explosive  mixture 
of  air  and  gasoline  vapor.    In  con- 
struction   and 
action  it  is  es- 
sentially    the 
same    as    the 
gas    engine. 
The  liquid  fuel 
is  vaporized  in 
a  device  called 

a     carbureter,  FlG-  209'~ A  Gasoline 

placed  near  the  inlet  valve  of  the  cylinder.  The  carbureter  thus 
corresponds  to  the  producer  of  the  producer-gas  engine. 

The  special  advantages  of  the  gasoline  engine  are  that  it  is  a  com- 
plete power  plant  in  itself,  is  small  and  light,  and  is  simple  and  con- 
venient to  operate.  It  supplies  the  power  for  motor  cycles,  motor 
boats,  gasoline  launches,  automobiles,  and  flying  machines,  and  is 
extensively  used  in  the  form  of  stationary  and  portable  engines  for 
running  light  machinery.  The  automobile  engine  usually  has  four 
cylinders,  sometimes  six,  operating  the  same  shaft.  The  power 
strokes  of  the  different  cylinders  occur  one  at  a  time;  hence  with  the 
four-cylinder  engine  there  is  one  power  stroke  to  each  half  turn  of 
the  shaft,  as  in  the  case  of  the  single-cylinder  steam-engine.  With  a 
six-cylinder  engine  there  are  three  power  strokes  to  each  revolution 
of  the  shaft,  and  the  motion  is  very  steady. 

257.  The  Steam  Turbine.  — All  engines  having  a  cylin- 
der and  piston  are  classed  as  reciprocating  engines,  from 
the  to-and-fro  or  reciprocating  motion  of  the  piston  and 
the  parts  intervening  between  it  and  the  crank-shaft.  The 
sudden  starting  and  stopping  of  these  parts  with  every 
stroke  causes  vibration  and  loss  of  power;  and  this  type  of 
motion  is  objectionable  in  other  respects. 


HEAT  ENGINES 


303 


These  difficulties  are  wholly  avoided  in  the  steam  tur- 
bine, in  which  the  only  moving  parts  are  a  revolving  wheel 
and  shaft.  The  wheel  of  a 
simple  steam  turbine  —  the  De 
Laval  —  is  shown  in  Fig.  210. 
It  carries  a  single  set  of  concave 
blades  upon  its  circumference, 
and  is  driven  by  jets  of  steam, 
issuing  from  nozzles  and 
directed  at  the  proper  angle 
against  the  blades.  In  the  illus- 
tration one  of  the  nozzles  is 
represented  as  transparent,  to 
show  the  diverging  outlet.  On  account  of  this  divergence 
the  steam  expands  in  passing  through  the  nozzle,  and  in 


FIG.  210.  —  De  Laval  Steam 
Turbine. 


FIG.  211.  —  De  Laval  Steam  Turbine  and  Dynamo. 

expanding  it  acquires  a  high  velocity,  its  potential  energy 
thus  becoming  kinetic.  The  velocity  of  the  steam  as  it 
strikes  the  blades  is  in  some  cases  as  high  as  4000  ft.  per 
second  or  45  mi.  per  minute.  The  wheel  rotates  at  speeds 
ranging  from  10,000  to  30,000  revolutions  per  minute. 


304  HEAT 

This  speed  is  too  great  for  direct  utilization,  and  is  reduced 
by  gear-wheels  in  the  ratio  of  10  to  i. 

The  De  Laval  steam  turbine  is  used  for  operating  dyna- 
mos, centrifugal  pumps,  blowers,  etc.  A  3o-horse-power 
turbine-dynamo  unit  is  shown  in  Fig.  211.  The  turbine 
wheel  is  inclosed  in  the  short  cylindrical  case  at  the  right. 
The  large  cylindrical  case  at  the  center 
incloses  the  gear-wheels,  by  which  the 
speed  is  reduced  in  transmitting  the 
motion  to  the  armature  shaft. 

By  means  of  a  multiple  turbine 
wheel  greater  power  can  be  developed 
and  at  slower  speeds.  The  Parsons 
and  the  Curtis  turbines  are  of  this 
type.  The  wheel  or  rotor  is  a  long  cyl- 
FIG.  212.  — Diagram  of  inder,  around  which  there  are  many 


in,,  Tuib-ne  rows  of  blades,  with  spaces  between 

Wheel.  M,  Revolving 

Blades;  s,  stationary  the  rows.     These  spaces  are  occupied 

by  rows  of  fixed  blades,  which  project 

inward  from  the   cylindrical  case.     The  arrangement   is 

shown  in  Fig.  212.     The  fixed  blades  serve  to  direct  the 

steam  against  the  successive  sets  of  moving  blades. 

The  steam  turbine  is  the  latest  type  of  heat  engine  to  achieve 
success.  Its  earliest  use  dates  from  1883.  Since  then  it  has  been 
greatly  improved  and  adapted  to  many  different  kinds  of  work.  The 
efficiency  of  the  earlier  turbines  was  very  low.  The  modern  turbine 
is  superior  to  the  best  reciprocating  steam-engines  in  this  respect. 
It  gives  the  best  results  when  operating  as  a  condensing  engine,  with 
superheated  steam  at  high  pressure.  The  steam  turbine  has  the 
further  advantage  of  being  much  smaller  than  a  reciprocating  engine 
of  the  same  power.  Many  steamships,  cruisers,  and  other  vessels 
built  in  recent  years  have  been  equipped  with  turbine  engines  of  the 
Parsons  and  the  Curtis  types;  and  electricity  for  lighting  and  power 
purposes  is  now  generated  in  large  centers  of  population  by  turbine- 
dynamo  units  of  5000  to  18,000  horse-power. 


CHAPTER  IX 
SOUND 

258.  Introduction. —The  five  senses  —  sight,  hearing, 
smell,  taste,  and  touch  —  are  the  channels  through  which 
the  mind  receives  impressions  from  the  outer  world.  These 
impressions  furnish  the  raw  materials  out  of  which  the  mind, 
by  processes  of  reasoning,  constructs  scientific  knowledge. 
Through  the  sense  of  touch  we  receive  certain  impressions 
by  which  we  know  whether  a,  body  is  hot  or  cold;  but  the 
nature  of  heat  was  discovered  only  by  reasoning,  based 
upon  experiment  and  a  general  knowledge  of  physical  prin- 
ciples. Through  the  sense  of  hearing  we  receive  a  class 
of  impressions  called  sensations  of  sound.  Experience 
teaches  that  these  sensations  are  due  to  vibrating  bodies, 
which  may  be  near  or  distant.  Obviously  some  invisible 
action  takes  place  across  the  intervening  space  between 
the  vibrating  body  and  the  ear.  This  action  we  call 
sound.  It  is  the  physical  cause  of  the  sensation,  but  its 
nature  is  not  revealed  by  ordinary  experience.  We  might 
hear  sounds  all  our  lives  without  learning  what  sound  is. 
We  describe  different  sounds  in  terms  of  the  sensations 
which  they  produce,  calling  them  musical,  unmusical, 
loud,  faint,  shrill,  high,  low,  deep,  sweet,  melodious,  hol- 
low, harsh,  discordant,  etc.  These  wonderfully  varied 
impressions  must  be  due  to  certain  physical  differences 
between  one  sound  and  another;  but  the  nature  of  these 
differences  and  the  nature  of  sound  itself  can  be  deter- 
mined only  by  reasoning  and  experiment. 

305 


3o6  SOUND 

In  the  study  of  sound  as  a  branch  of  physics,  we  are 
principally  concerned  with  such  questions  as  these,  which 
relate  to  physical  processes  rather  than  to  the  sensations 
produced  by  them.  It  should  be  noted  at  the  outset  that 
these  processes  are  mechanical,  and  are  in  perfect  agree- 
ment with  mechanical  laws  and  principles. 

I.    ORIGIN  AND  TRANSMISSION  OF  SOUND 

259.  Sounding  Bodies.  —  It  can  be  shown  in  various 
ways  that  a  sounding  body  is  in  a  state  of  vibration.  In 
many  cases  this  is  evident  from  the  appearance  of  the  body, 
although  the  motion  is  always  too 
rapid  to  follow  with  the  eye.  A 
string  of  a  musical  instrument  has 
the  appearance  of  a  gauzy  spindle 
when  sounding;  and  the  prongs  of 
a  tuning  fork  become  indistinct  and 
appear  to  widen  out  at  the  free 
ends,  where  the  motion  is  greatest. 
The  vibrations  can  generally  be  felt, 
and,  unless  very  weak,  can  be  shown 

1.3.-  Vibrating  spring.    by    ^    mechankal     effectg         For 

example,  a  shower  of  spray  is  thrown  up  when  the  prongs 
of  a  vibrating  fork  are  dipped  into  water;  and  a  pith  ball 
or  a  bit  of  cork,  tied  to  the  end  of  a  thread,  is  driven  away 
when  suspended  so  as  to  touch  the  edge  of  a  sounding  bell. 
Sound  may  also  be  produced  by  a  vibrating  body  of  liquid 
or  gas.  The  sound  of  running  water  and  the  notes  of  wind 
instruments  are  familiar  examples. 

Sounding  bodies  differ  from  one  another  in  their  modes 
of  vibration,  and  the  same  body  may  vibrate  differently 
under  different  conditions.  One  of  the  simplest  cases  is 
that  of  a  long,  straight,  steel  spring,  rigidly  fastened  at  a 


ORIGIN  AND  TRANSMISSION  OF  SOUND         307 

greater  or  less  distance  from  the  vibrating  end  (Fig.  213). 
As  this  end  is  gradually  shortened  the  vibrations  become 
more  and  more  rapid,  until,  at  a  certain  rate,  a  low,  musical 
note  is  heard.  Incidentally  we  may  observe  that  the  spring 
sounds  a  higher  note  as  its  length  is  further  decreased, 
illustrating  the  fact  that  the  pitch  of  a  sound  is  determined 
by  the  rate  of  vibration  of  the  sounding  body.  This  im- 
portant relation  will  be  studied  later.  Our  present  purpose 
is  to  observe  the  manner  in  which  the  spring  vibrates. 
The  character  of  the  vibration  remains  the  same  whether 
it  is  slow  or  rapid;  it  can  therefore  be  studied  by  adjusting 
the  spring  to  one  or  two  swings  per  second.  It  will  then 
be  seen  that  the  motion  is  similar  to  that  of  a  pendulum. 
During  the  first  half  of  each  swing  the  motion  is  acceler- 
ated, during  the  latter  half  it  is  retarded;  it  is  most  rapid 
at  the  mid-position  D.  The  vibratory  motion  is  main- 
tained by  the  elastic  force  of  the  spring,  which  plays  the 
same  part  as  the  force  of  gravity  upon  a  pendulum  bob. 
In  the  study  of  sound,  one  vibration  includes  a  swing  both 
ways  (from  D'  to  D"  and  back  to  Df).  The  amplitude  of 
vibration  is  the  extent  of  motion  on  either  side  of  the  posi- 
tion of  rest.  The  rate  of  vibration  of  sounding  bodies  is 
independent  of  the  amplitude;  the  vibration  is  therefore 
regular  or  periodic.  This  can  be  shown  in  the  case  of  the 
steel  spring  by  counting  the  number  of  vibrations  in  a 
given  time,  with  different  amplitudes;  but  it  is  proved  for 
all  sounding  bodies  by  the  fact  that  a  sound  does  not  change 
in  pitch  as  it  becomes  fainter  and  gradually  dies  away. 
The  rate  of  vibration  is  measured  by  the  number  of  vibra- 
tions per  second. 

Tuning  forks  are  so  frequently  used  in  experiments  in  sound  that 
it  is  important  to  know  how  a  fork  vibrates.  A  quick  blow  upon  one 
prong,  in  the  direction  of  the  other,  sets  both  prongs  in  vibration. 


308 


SOUND 


Their  motion  is  always  toward  each  other  and  from  each 
other  in  succession  (Fig.  214).  The  transverse  vibration 
of  the  prongs  is  accompanied  by  a  vibration  of  the  stem 
in  the  direction  of  its  length  (longitudinal  vibration) ,  which 
can  be  distinctly  felt  by  placing  the  stem  of  a  sounding 
fork  against  the  teeth.  A  sounding  fork  is  always  held  or 
supported  by  the  stem;  touching  a  prong  quickly  stops  it. 


260.  Sound  Media. — In  ordinary  circumstances  FlG  2I4 
sound  is  conveyed  to  the  ear  through  the  air.  Does 
the  air  play  an  essential  part  in  the  process?  We  know 
that  radiant  energy  is  transmitted  through  the  air,  but  not 
by  means  of  it.  A  vacuum  serves 
the  purpose  quite  as  well  or  even 
better  (Art.  194).  That  the  air  does 
take  an  active  and  necessary  part 
in  the  transmission  of  sound  is  read- 
ily shown  with  an  electric  bell,  stand- 
ing on  a  soft  cushion  or  suspended 
by  fine  wires  within  the  receiver  of 
an  air  pump  (Fig.  215).  Before  the 
air  is  exhausted  the  bell  can  be  heard 
distinctly;  but  the  sound  grows  con- 
tinually fainter  as  the  exhaustion  proceeds,  and  ceases 
when  a  good  vacuum  is  secured.  It  is  restored  when  air 
or  any  other  gas  is  admitted  into  the  receiver. 

Sound  is  also  transmitted  by  liquids  and  solids.  Its 
transmission  through  the  walls  and  floors  of  buildings  is 
familiar,  and  the  faintest  tapping  or  scratching  at  one  end 
of  a  long  board  or  table  can  be  heard  at  the  farther  end 
wnen  the  ear  is  pressed  against  it.  A  swimmer,  with  his 
head  under  water,  hears  a  loud  sound  when  two  stones 
are  struck  together,  also  under  water,  at  a  long  distance 
from  him. 


FIG.  215. 


ORIGIN  AND  TRANSMISSION  OF  SOUND         309 

Whether  all  solids  transmit  sound  equally  well  can  be 
readily  determined  by  experiment.  When  the  stem  of  a 
sounding  fork  is  pressed  against  the  top  of  a  table,  the  sound 
becomes  much  louder.  The  stem  in  vibrating  strikes  the 
table  with  a  rapid  succession  of  blows,  and  the  impulses 
thus  imparted  cause  the  table  to  vibrate  in  unison  with 
the  fork.  The  table  thus  becomes  a  sounding  body.  The 
sound  from  the  table  is  almost  if  not  quite  as  loud  when  con- 
nection between  it  and  the  sounding  fork  is  made  through  a 
meter  stick,  or  other  long  rod  of  wood  or  metal;  but  little 
or  no  sound  is  heard  when  a  rubber  stopper  or  a  roll  of  cot- 
ton-wool is  placed  between  the  fork  and  the  table.  All 
highly  elastic  (rigid)  bodies  transmit  sound  well;  soft  and 
yielding  solids  transmit  it  poorly.  The  latter  are  said  to 
"deaden"  the  sound. 

From  such  experiments  as  the  above  we  learn  that  sound 
can  exist  only  in  ordinary  matter,  and  that  its  transmis- 
sion depends  upon  the  elasticity  of  the  substance  through 
which  it  is  passing.  Any  substance  that  transmits  sound 
is  called  a  sound  medium  (plural,  media). 

261.  Wave  Motion.  —  Taking  together  the  facts  that 
sound  originates  at  vibrating  bodies  and  that  it  is  trans- 
mitted by  the  surrounding  air  and  other  elastic  media, 
the  inference  is  plain  that  it  must  be  some  kind  of  dis- 
turbance produced  in  the  medium  itself.  Since  this  disturb- 
ance is  invisible,  we  can  better  understand  what  it  is  like, 
how  it  is  produced,  and  how  it  is  propagated  after  a  pre- 
liminary study  of  a  few  types  of  visible  motion  which  are 
in  some  respects  like  it. 

When  a  stretched  rubber  tube  or  a  spiral  spring,  three  or 
four  meters  long,  is  struck  a  sharp  blow  near  one  end,  a 
distortion  is  produced  which  travels  rapidly  as  a  wave  to 


310  SOUND 

the  other  end.  By  tying  strips  of  cloth  to  the  tube  at 
different  points,  it  can  be  seen  that,  as  the  wave  passes 

any  point,  that  point 
moves  quickly  out  in  a 
direction  at  right  angles 

FIG.  216.  — Longitudinal  Vibration  in  a         to     the     length     of      the 
Spiral  Spring.  ,11  r™ 

tube  and  returns.  The 

curved  form  that  we  call  the  wave  is,  in  fact,  passed  from 
point  to  point  along  the  tube  by  the  transverse  vibration  of 
successive  portions  of  the  tube.  A  distortion  of  a  differ- 
ent character  is  started  by  stretching  a  portion  of  the  tube 
near  one  end  either  considerably  more  or  less  than  the 
remainder  and  suddenly  releasing  that  portion  (Fig.  216). 
The  strips  of  cloth  now  indicate  a  to-and-fro  or  longitu- 
dinal vibration  as  the  disturbance  passes. 

The  waves  that  travel  over  a  field  of  grain  when  dis- 
turbed by  the  wind  are  formed  of  the  swaying  stalks,  as 
they  bend  forward  and  spring  back.  The  wave  form  pro- 
duced by  this  motion  of  the  stalks  sweeps  over  the  field 
in  the  direction  of  the  wind. 

A  water  wave  is  transmitted  by  a  vibratory  motion  of 
the  water  particles.  This  is  mainly  an  upward  and  down- 
ward or  transverse  vibration,  as  is  shown  by  the  rise  and 
fall  of  any  floating  object  as  the  waves  pass  under  it.  A 
wave  is  not  formed  of  the  same  water  as  it  travels ;  it  is  the 
disturbance  that  travels,  not  the  water.  A  pebble  dropped 
into  a  pool  produces  a  train  of  circular  waves,  which 
travel  outward  from  the  point  where  the  pebble  strikes. 
The  waves  are  circular  because  the  disturbance  is  trans- 
mitted with  equal  velocity  in  all  directions  over  the  sur- 
face; they  are  concentric  because  they  all  start  at  the 
same  point.  Each  wave  consists  of  a  crest  and  a  trough. 
The  length  of  a  wave  is  the  distance  from  crest  to  adja- 


ORIGIN  AND  TRANSMISSION  OF  SOUND        311 

cent  crest  or  from  trough  to  adjacent  trough,  measured 
at  right  angles  to  the  line  of  the  crest.  The  waves  rapidly 
decrease  in  height  as  they  travel  because  their  energy  is 
handed  on  to  an  ever-increasing  body  of  water. 

A  wave  of  any  sort  is  made  up  of  moving  and  distorted 
parts  of  the  medium  in  which  it  is  traveling,  and  hence 
possesses  energy.  This  energy  goes  with  the  wave,  the 
net  result  of  the  process  being  a  transmission  of  energy, 
but  not  of  matter. 

262.  Sound  Waves.  —  If  the  action  upon  the  air  were 
the  same  on  all  sides  of  a  sounding  body  at  the  same  in- 
stant, a  sound  wave  would  have  the  form  of  a  hollow  sphere, 
with  the  sounding  body  at  its  center.  Generally  the  waves 
are  only  greater  or  less  fractions  of  such  spherical  shells, 
depending  upon 'the  particular  manner  in  which  the  sound- 
ing body  vibrates;  but  the  simplest  case  is  that  of  a  com- 
plete spherical  wave,  such  as  is  produced  by  the  explosion 
of  a  firecracker  in  the  air.  Let  us  try  to  form  a  mental 
picture  of  such  a  wave.  The  gas  generated  by  the  explo- 
sion of  the  powder  in  the  firecracker  instantly  expands, 
forcing  the  air  violently  away  on  all  sides.  This  shell 
of  outward  moving  air,  on  account  of  its  momentum, 
goes  farther  than  is  necessary  to  restore  equilibrium,  leav- 
ing a  partial  vacuum  at  the  center  of  disturbance.  There 
is  now  an  unbalanced  pressure  toward  the  center,  which 
drives  the  air  back  again.  While  the  air  particles  are 
moving  outward  they  form  a  spherical  shell  of  increased 
density,  represented  by  CiC2C3C4  (Fig.  217).  During  their 
inward  motion  they  form  a  shell  of  diminished  density, 
represented  by  RiR2R3Rt.  The  former  is  called  a  compres- 
sion, the  latter  a  rarefaction.  The  two  together  consti- 
tute a  sound  wave.  The  air  particles  in  the  compression 


312  SOUND 

transmit  their  forward  motion  to  the  particles  just  ahead, 
and  themselves  come  to  rest.  The  compression  thus  travels 
outward  through  the  air.  At  the  rear  (inside)  of  the  com- 
pression the  forward  motion  of  the  particles  has  just  ceased, 
and  they  are  on  the  point  of  moving  backward  into  the 


FIG.  217.  —  Section  of  a  Sound  Wave  in  Air. 

rarefaction.  This  backward  motion  of  the  air  particles 
transmits  the  rarefaction  forward,  following  the  compres- 
sion. The  forward  and  backward  motion  (longitudinal 
vibration)  of  the  particles  of  the  medium  should  be  care- 
fully distinguished  from  the  steady  onward  motion  of  the 
wave  through  the  medium. 

The  mechanical  action  of  the  air  particles  in  transmitting  a  sound 
wave  is  similar  to  that  of  a  row  of  balls  in  transmitting  an  impulse 
from  one  to  another  along  the  row.  This  action  can  be  shown  with 
.a  number  of  balls,  placed  close  together  in  a  groove  (Fig.  218).  When 
a  ball  is  rolled  against  one  end  of  the  row,  each  one  in  the  row  collides 
with  the  next  in  rapid  succession,  and  the  last  one  is  driven  away. 
The  motion  of  the  others  is  scarcely  perceptible,  although  each  in 


ORIGIN  AND  TRANSMISSION  OF  SOUND         313 

turn  receives  and  transmits  the  energy  that  is  imparted  finally  to 
the  one  at  the  end.    It  should  be  noted  that  the  last  ball  flies  away 
seemingly    at    the    very 
instant    when    the    first 
one  is  struck.    The  speed 
of     the    disturbance     is 
many  times  greater  than 
that  of  any  of  the  balls. 


FIG.  218. 


263.  A  Musical  Sound  consists  of  a  continuous  series 
or  train  of  waves  of  the  same  character.  The  sound  of 
a  tuning  fork  is  a  good  example.  As  either  prong  of  a 
fork  advances,  it  drives  the  air  rapidly  before  it,  producing 
a  compression  (Fig.  219,  I  and  III);  as  it  retreats,  the  air 
follows  it  up,  producing  a  rarefaction  (II  and  IV  in  the 
figure).  A  wave  is  thus  produced  by  each  complete  vibra- 
tion. (The  waves  spread  out  as  they  travel.  The  sec- 
tions represented  in  the  figure  are  only  partial.) 


FIG.  219. —  Propagation  of  Sound  Waves. 


The  air  particles  at  the  fork  vibrate  with  the  amplitude  and  velo- 
city of  the  prong.  This  amplitude  is  rarely  more  than  i  mm.,  and  it 
grows  less  at  increasing  distances  from  the  fork,  owing  to  the  increase 
in  the  size  of  the  waves.  At  the  fork  the  air  particles  move  to  and  fro 
with  the  prong;  but  as  the  distance  from  the  fork  increases  the  vi- 
brations of  the  particles  lag  more  and  more  behind,  owing  to  the  fact 


314  '  SOUND 

that  it  takes  time  for  the  disturbance  to  travel.  This  can  be  shown 
with  the  aid  of  the  figure.  While  the  prong  is  making  its  first  out- 
ward swing  the  disturbance  advances  to  a\  in  I,  a  point  at  a  distance 
of  about  two  feet  if  the  pitch  ot  the  fork  is  "middle  C."  The  prong 
and  the  adjacent  air  particles  at  c\  complete  their  forward  motion  at 
the  same  instant.  The  particles  at  b\  are  then  at  the  mid-point  of 
their  forward  swing,  and  those  at  a\  are  just  starting.  During  the 
backward  swing  of  the  prong,  the  front  of  the  compression  advances 
to  az  in  II,  and  its  rear  to  cz.  Meanwhile  the  retreat  of  the  prong 
has  produced  a  rarefaction,  which  extends  from  ez  to  a.  At  ez  the 
backward  swing  of  the  air  particles  has  just  ceased,  with  that  of  the 
fork;  at  dz  it  is  half  completed;  at  cz  it  is  just  beginning.  Parts 
III  and  IV  of  the  figure  represent  the  state  of  things  at  the  end  of 
the  second  forward  and  the  second  backward  swing  of  the  prong 
respectively. 

A  sound  wave  consists  of  a  compression  and  the  follow- 
ing rarefaction  A  wave  front  is  the  surface  bounding 
the  front  of  a  compression.  The  length  of  a  sound  wave 
is  the  distance  from  wave  front  to  wave  front.  A  wave 
travels  its  own  length  while  the  sounding  body  is  making 
one  complete  vibration;  and  its  length  remains  the  same 
however  far  it  may  travel. 

264.  Intensity  and  Loudness  of  Sounds.  —  The  inten- 
sity of  a  sound  refers  to  the  mechanical  energy  of  the  waves. 
It  is  a  physical  quantity.  The  loudness  of  a  sound  refers 
to  the  sensation  produced  when  the  sound  is  heard.  It 
depends  in  part  upon  the  condition  of  the  ear  and  in  part 
upon  the  pitch  of  the  sound,  a  shrill  sound  being  more  dis- 
tinctly heard  than  one  of  equal  intensity  but  of  low  pitch. 
In  general,  however,  the  loudness  of  a  sound  increases 
with  its  intensity. 

In  physics  we  are  concerned  with  the  conditions  which 
determine  the  intensity  of  a  sound  at  its  source  and  its 
loss  of  intensity  in  transmission. 


ORIGIN  AND  TRANSMISSION  OF  SOUND         315 

265.  The  Intensity  at  the  Source  is  proportional  to  the 
rate  at  which  the  air  or  other  medium  receives  energy 
from  the  sounding  body;  and  this  depends  upon  the  ampli- 
tude of  vibration  of  the  body,  the  area  of  the  vibrating 
surface,  and  the  density  and  elasticity  of  the  medium. 

Effect  of  Amplitude.  —  The  gradual  dying  away  of  the 
sound  of  a  bell,  a  piano  wire,  a  tuning  fork,  etc.,  is  due  to 
the  diminishing  amplitude  of  vibration  as  the  body  ap- 
proaches a  state  of  rest.  With  a  decrease  of  amplitude  the 
blows  of  the  vibrating  surface  against  the  surrounding  air 
grow  less  vigorous,  and  the  sound  waves  are  correspondingly 
weaker. 

Effect  of  the  Area  of  the  Vibrating  Surface.  —  A  narrow 
vibrating  surface  cuts  through  the  air,  producing  little 
effect;  the  air  slips  round  it.  A  broad  surface  catches  the 
air  and  carries  it  bodily  forward.  This  is  the  reason  why 
the  sound  of  a  tuning  fork  is  very  faint  when  it  is  held 
in  the  hand  and  loud  when  it  is  touched  to  a  table.  In 
the  latter  case  the  vibrations  are  transmitted  to  the  air 
almost  entirely  by  the  vibrating  table.  The  music  of  a 
violin  or  guitar  comes  practically  entirely  from  the  body 
of  the  instrument,  and  the  music  of  a  piano  from  the  sound- 
ing board  on  which  the  wires  are  strung.  (The  bodies  of 
stringed  instruments  are  capable  of  taking  up  the  vibra- 
tions of  any  number  of  the  strings  at  the  same  time.) 

Effect  of  the  Density  and  Elasticity  of  the  Medium.— In 
the  experiment  with  the  sounding  body  under  the  receiver 
of  an  air  pump,  it  was  observed  that  the  sound  grows  fainter 
as  the  exhaustion  continues,  i.e.  as  the  density  of  the  remain- 
ing air  is  diminished.  The  reason  is  obvious:  there  is  less 
matter  in  motion  in  the  wave  of  rarefied  air,  hence  there 
is  less  energy  —  kinetic  energy  being  proportional  to  the 


SOUND 

mass  of  the  moving  body.  "On  high  mountains,  where 
the  air  is  much  rarefied,  it  is  necessary  to  speak  with  some 
effort  in  order  to  be  heard,  and  the  discharge  of  a  gun 
produces  only  a  feeble  sound. " 

It  has  already  been  shown  in  a  number  of  experiments  that  sound 
is  louder  when  transmitted  through  elastic  solids  than  it  is  when  trans- 
mitted through  the  air.  We  are  not  to  infer  that  solids  are  better 
transmitters  of  sound  than  air  is,  but  rather  that  the  intensity  of  the 
sound  is  greater  at  the  source  when  the  medium  is  a  solid.  For 
example,  a  vibrating  fork  produces  waves  of  greater  intensity  in  the 
top  of  a  table  than  it  does  in  the  air  directly;  for  the  rigid  wood  offers 
much  greater  resistance  to  the  blows  of  the  fork  than  the  air  does, 
and  consequently  receives  a  greater  amount  of  energy  with  each 
vibration. 

266.  Loss  of  Intensity  in  Transmission.  —  It  is  well 
known  that  a  sound  grows  fainter  with  increasing  distance 

from  its  source.  The 
definite  relation  be- 
tween intensity  and  dis- 
tance follows  from  the 
principle  of  the  conser- 
vation of  energy,  and 
holds  for  light  as  well 
as  for  sound.  This  im- 
portant law  is  derived 
as  follows: 

As  a  sound  wave  travels  in  the  open  air,  its  distance 
from  the  source  is  the  radius  of  its  surface  (the  spherical 
wave  front).  Now  it  is  proved  in  geometry  that  the  sur- 
faces of  two  spheres  (or  equal  fractions  of  their  respective 
surfaces)  are  proportional  to  the  squares  of  their  radii. 
Hence,  since  the  length  of  a  wave  remains  constant,  its 
volume  and  the  mass  of  air  composing  it  are  proportional 
to  the  square  of  the  distance  it  has  traveled. 


FIG.  220. —Relation  of  Volume  to  Distance. 


ORIGIN  AND  TRANSMISSION  OF  SOUND        317 

This  is  shown  in  Fig.  220  which  represents  a  section  of  a  spherical 
wave  at  a  distance  d\  from  the  source  and  again  at  twice  that  dis- 
tance, or  fa.  If  vi  denotes  the  volume  of  the  section  at  the  first 
distance,  and  %  its  volume  at  the  second  distance,  then  vz  :  vi  ::  4  :  i, 
or,  in  general,  vz  :vi  ::  ck2  :  d?. 

Assuming  that  the  total  energy  of  a  sound  wave  remains 
unchanged  as  the  wave  travels,  the  intensity  of  the  sound 
(or  the  amount  of  energy  per  unit  volume  of  the  medium) 
varies  inversely  as  the  volume  of  the  wave,  and  hence,  in 
the  open  air,  it  varies  inversely  as  the  square  of  the  distance 
from  the  source. 

This  relation  is  not  strictly  true,  for  the  energy  of  sound 
is  more  or  less  slowly  transformed  into  heat  by  internal 
friction  in  any  medium.  In  the  end  it  is  all  dissipated 
in  this  manner.  Hence  the  intensity  decreases  somewhat 
more  rapidly  than  the  law  indicates. 

267.  Confined  Sound  Waves.  —  Sound  travels  long  distances  in 
elastic  media  with  very  little  loss  of  intensity  when  the  waves  are 
prevented  from  increasing  in  size.    This  is  the  principle  of  the  speak- 
ing tube,  which  is  a  long  metal  tube  of  small  diameter,  used  for 
Communication  between  different  rooms  of  a  building  or  between 
some  room  and  the  street  door.     The  tube  does  not  readily  take  up 
the  vibrations  of  the  air,  hence  the  sound  is  almost  completely  con- 
fined within  it.     Practically  the  only  loss  is  that  due  to  friction. 

Solids,  such  as  rods,  stretched  wires  and  strings,  the  rails  of  a 
track,  etc.,  transmit  sound  with  comparatively  little  loss  of  intensity 
for  the  same  reason,  the  vibrations  being  largely  confined  in  the  solid 
medium.  This  fact  is  utilized  in  the  acoustic  or  string  telephone, 
which  is  capable  of  rendering  practical  service  over  short  distances. 

268.  The  Velocity  of  Sound  in  Air.  —  It  is  a  matter  of 
common  observation  that  an  event  which  takes  place  at 
a  distance  is  seen  before  the  sound  produced  by  it  is  heard. 
At  a  distance  of  a  few  hundred  feet  the  blow  of  a  hammer 
is  heard  after  the  hammer  is  raised  for  the  next  stroke; 


3i8  SOUND 

the  cloud  of  "steam"  issuing  from  the  whistle  of  a  dis- 
tant locomotive  is  seen  and  may  even  disappear  before 
the  whistle  is  heard;  the  thunder  that  accompanies  a  flash 
of  lightning  is  often  delayed  many  seconds  in  reaching  the 
ear.  Now  the  time  required  for  light  to  travel  terres- 
trial distances  is  inappreciable  (the  velocity  of  light  being 
186,000  mi.  per  second);  hence  the  interval  between  the 
sensations  of  sight  and  hearing  in  such  phenomena  as 
these  is  the  time  occupied  by  the  sound  in  traveling  from 
the  sounding  body  to  the  observer,  and  if  the  time  and  the 
distance  are  known  the  velocity  of  sound  can  be  computed. 

Observations  have  repeatedly  been  taken  for  this  purpose 
by  firing  a  cannon  at  each  of  two  stations  several  miles 
apart,  and  noting  the  time  between  the  flash  and  the  re- 
port as  observed  at  the  other  station.  By  taking  observa- 
tions at  each  of  the  stations  alternately,  a  correction  can 
be  made  for  the  effect  of  the  wind.  The  average  of  the 
best  determinations  is  332  m.  or  1090  ft.  per  second  at 
o°  C.  At  20°  the  velocity  is  344  m.  or  1129  ft.  per  second. 

That  the  velocity  of  sound  is  independent  of  its  pitch 
and  intensity  is  proved  by  the  fact  that  all  notes  sounded 
together  by  an  orchestra  are  heard  together  at  all  distances. 

269.   Causes  which  Determine  the  Velocity  of  Sound.  - 

The  transmission  of  waves  of  any  kind  in  ordinary  matter 
is  a  mechanical  process,  and  is  in  agreement  with  the 
general  laws  of  dynamics.  The  mathematical  physicist, 
starting  with  these  general  laws  and  the  physical  proper- 
ties of  gases,  can  compute  the  velocity  of  sound  in  air  with- 
out resorting  to  experiment ;  but  the  factors  upon  which  the 
velocity  of  a  wave  depends  can  very  well  be  shown  in  a 
general  way  by  means  of  two  long  rubber  tubes  of  the  same 
size,  one  of  them  filled  with  shot  or  sand.  With  the  empty 


ORIGIN  AND  TRANSMISSION  OF  SOUND         319 

tube,  a  wave  started  by  striking  it  sharply  near  one  end 
travels  rapidly  back  and  forth  from  end  to  end,  the  speed 
increasing  with  the  tension  of  the  tube.  In  transmitting 
the  wave,  successive  portions  of  the  tube  swing  out  to  one 
side  and  are  jerked  back  again  by  the  elastic  resistance 
developed  in  the  stretched  rubber.  This  transverse  vibra- 
tion is  quickened  by  increasing  the  tension,  in  accordance 
with  the  law  that  the  acceleration  of  a  given  mass  is  pro- 
portional to  the  force.  When  the  weighted  tube  is 
stretched  equally  with  the  other,  the  waves  travel  much 
more  slowly  in  it,  according  to  the  law  that,  with  a  given 
force,  the  acceleration  varies  inversely  with  the  mass. 

Thus  in  general  the  two  factors  which  control  the  trans- 
mission of  a  disturbance  in  any  medium  are  force  and 
mass.  In  the  case  of  sound  waves  the  force  is  due  to  the 
elasticity  of  volume  of  the  medium  (expansive  force),  and 
the  other  factor  is  the  mass  per  unit  volume,  or  the  den- 
sity. The  exact  relation  is  that  the  velocity  varies  directly 
as  the  square  root  of  the  elasticity  and  inversely  as  the 
square  root  of  the  density.  The  velocity  of  sound  in  water 
has  been  found  by  experiment  to  be  1435  m.  per  second 
at  8°,  a  velocity  more  than  four  times  as  great  as  in  air. 
Thus,  in  comparison  with  air,  the  retarding  effect  of  the 
greater  density  of  water  is  more  than  offset  by  the  acceler- 
ating effect  of  its  still  greater  relative  elasticity  (rigidity) . 
This  is  true  in  even  greater  degree  of  solids,  the  velocity  in 
glass  and  steel  being  about  fifteen  times  as  great  as  in  air. 

The  velocity  of  sound  in  air  increases  with  the  temperature,  be- 
cause air  expands  with  a  rise  of  temperature,  and  its  density  diminishes 
while  its  elasticity  remains  unchanged.  The  gain  of  velocity  is  i 
ft.  per  second  per  degree  Fahrenheit,  or  .6  m.  per  second  per 
degree  Centigrade.  An  increase  of  pressure  increases  the  elasticity 
and  the  density  in  the  same  ratio;  hence  a  change  of  barometric 
pressure  does  not  affect  the  velocity. 


320 


SOUND 


270.  Reflection  of  Sound.  Echoes.  — When  sound  waves 
strike  a  large  surface,  as  a  cliff  or  the  side  of  a  building, 

they  are  reflected.  The  re- 
flected sound  is  called  an 
echo  when  it  reaches  the 
ear  long  enough  after  the 
original  sound  to  be  distin- 

Suisned  from  it:  This  re- 
quires about  one  fifth  of  a 

second  at  least,    during 
which    time   sound    travels 
68  m.  Hence  a  distinct  echo 
FIG.  221.  — Reflection  of  Sound  Waves  will  not  be  heard  unless  the 

from  a  Plane  Surface.  ~  ... 

reflecting   surface    is   at   a 

distance  of  34  m.  or  more  from  the  source  of  the  sound. 
At  less  distances  the  direct  and  the  reflected  sounds 
blend  more  or  less  perfectly  together.  When  the  reflect- 
ing surface  is  within  a  few  meters  of  the  source,  the 
two  sounds  are  heard  as  one,  and  the  only  effect  of 
the  reflection  is  a  greater  intensity.  A  good  example  is  the 
reflecting  surface  at  the  rear  of  a  band  stand.  When 
the  distance  is  nearly  great  enough  for  an  echo,  the  direct 
and  the  reflected  sound  are  mixed  confusedly,  causing 
indistinctness.  This  is  often  noticeable  in  large  halls. 

When  sound  waves  are  reflected  from  a  vertical  plane  surface, 
their  curvature  is  reversed  as  shown  in  Fig.  221,  in  which  0  is  the 
source,  AB  the  reflecting  surface,  and  0'  the  center  of  the  reflected 
waves.  The  reflected  waves  behave  in  all  respects  as  if  0'  were  their 
real  source.  They  increase  in  size  and  decrease  in  intensity  in  pro- 
portion to  the  square  of  their  distance  from  0' '.  The  energy  trans- 
mitted along  any  radius  from  O  is  transmitted  after  reflection  along 
the  corresponding  radius  from  O',  as  OA  and  AD,  OB  and  BE,  etc. 

After  reflection  from  a  large  concave  surface,  sound  waves  increase 
less  rapidly  in  size  and  decrease  less  rapidly  in  intensity  than  they 


ORIGIN  AND  TRANSMISSION  OF  SOUND         321 

do  from  a  plane  surface.  The  behavior  of  the  reflected  sound  in 
such  cases  is  similar  to  that  of  light  when  reflected  in  a  strong  beam 
from  the  concave  mirror  of  the  headlight  of  a  locomotive  or  a  street 
car.  To  secure'  this  effect  the  reflecting  walls  at  the  rear  of  band 


FIG.  222.  —  Sound  Wave   Reflected   from   a   Concave   Surface,   MN . 
source  of  the  sound;   B,   the  sound  focus. 


A,   the 


stands  are  made  concave.  When  a  sounding  body  is  beyond  a  certain 
distance  from  a  concave  surface,  the  reflected  waves  are  concave 
(Fig.  222).  Such  waves  decrease  in  size  and  increase  in  intensity 
as  they  travel  toward  a  point.  This  is  the  principle  of  "whispering 
galleries,"  which  owe  their  remarkable  effects  to  curved  walls  or 
ceilings.  "  Sails  of  ships  are  sometimes  inflated  by  the  wind  so  that 
they  act  as  concentrating  reflectors  of  sound.  Arnott  says  that  in 
coasting  off  Brazil  he  heard  the  bells  of  San  Salvador  from  a  distance 
of  no  mi.,  by  standing  before  the  mainsail,  which  happened  at  the 
time  to  assume  the  form  of  a  concave  reflector,  focusing  at  his  ear." 

PROBLEMS 

1.  By  what  force  are  waves  transmitted  along  a  stretched  rubber 
tube?     By  what  force  are  water  waves  transmitted? 

2.  Sound  waves  consist  of  compressions  and  rarefactions  in  all  media, 
solid,  liquid,  and  gaseous.     Is  their  transmission  due  to  the  elasticity  of 
form  or  volume  of  the  medium? 

3.  Would  a  sounding  body  continue  to  vibrate  longer  in  water  or  in 
air?   in  the  air  or  in  a  vacuum?     Why?     Find  by  trial  whether  a  tuning 
fork  vibrates  longer  when  held  in  the  hand  or  when  its  stem  is  in  contact 
with  a  table.     Explain. 


322  SOUND 

4.  How  does  the  intensity  of  sound  at  a  distance  of  5  m.  from  the 
source  compare  with  its  intensity  at  10  m.?     at  15  m.?   at  20  m.? 

5.  At  what  distance  is  the  intensity  of  sound  one  fourth  as  great  as 
at  100  m.?   one  half  as  great? 

6.  How  would    music    be    affected    if  sounds  of  different    pitch  or 
different  intensity  traveled  with  different  velocities? 

7.  If  a  cannon  ball  goes  6  mi.  at  an  average  speed  of   2500  ft.  per 
second,   how  does  its  time  of  flight  compare  with   the  time  required  for 
the  sound  of  the  firing  to  travel  the  same  distance  in  air  at  20°? 

8.  Referring   to   Fig.  221,   account   for   the   reversal  of  curvature  of 
sound  waves  when  reflected  from  a  plane  surface.     Where  is  the  center  of 
the  reflected  waves  with  respect  to  the  reflecting  surface  and  the  position 
of  the  source?     Account  for  this. 

9.  A  rifle  is  fired  on  one  side  of  a  canon  and  3.2  sec.  later  the  echo 
is  heard  from  the  opposite  side.     The  temperature  is  20°.     What  is  the 
width  of  the  canon? 

10.  A  flash  of  lightning  is  seen  12.5  sec.  before  the  thunder  is  heard. 
At  what  distance  did  the  lightning  occur,  the  temperature  being  20°? 

11.  The  mean  distance  of  the  sun  from  the  earth  is  93,0x30,000  mi. 
How  long  after  an  explosion  occurs  upon  the  sun  would  we  hear  it  if  air 
at  o°  were  provided  as  a  medium  for  the  transmission?     (Light  reaches  us 
from  the  sun  in  499  sec.) 

II.  PROPERTIES  OF  MUSICAL  SOUNDS 

271.  Musical  Sounds  and  Noises.  —  All  sounds  may  be 
classed  as  musical  sounds  and  noises,  although  the  divid- 
ing line  between  the  two  classes  is  rather  indefinite.  We 
distinguish  a  great  variety  of  musical  sounds  and  a  still 
greater  variety  of  noises,  and  employ  a  considerable  vocab- 
ulary of  adjectives  in  attempting  to  describe  them.  But 
with  all  their  variety  these  descriptive  terms,  at  least  so 
far  as  they  relate  to  musical  sounds,  may  be  grouped  under 
three  heads,  namely: 

1.  Loudness,  as  loud,  soft,  faint,  powerful,  weak; 

2.  Pitch,  as  high,  shrill,  piercing,  low,  deep,  grave; 

3.  Quality,  or  character  (in  the  narrower  sense),  as  melo- 
dious, harsh,  nasal,  hollow,  rich,  full,  sweet,  mellow,  dis- 


PROPERTIES   OF   MUSICAL  SOUNDS 


323 


,  cordant.  Language  fails  to  express  the  subtle  variations 
of  quality  which  the  ear  easily  recognizes.  It  is  by  differ- 
ence in  quality  that  we  distinguish  the  notes  of  one  musical 
instrument  from  those  of  another,  or  the  sound  of  a  familiar 
voice  from  a  thousand  others. 

Such  terms  as  these  evidently  refer  to  the  impressions 
which  different  sounds  produce  upon  the  hearer;  they 
describe  sensations,  not 
their  physical  cause. 
It  is  clear,  however, 
that  the  different  sen- 
sations must  be  due  to 
differences  of  some  sort 
between  the  sound 
waves  themselves;  and 
it  is  with  these  char- 
acteristics of  the 
waves  and  the  man- 
ner in  which  they  are 
produced  by  the  sound- 
ing body  that  we  are  directly  concerned.  The  first  point 
to  be  determined  is  the  difference  between  a  musical  sound 
and  a  noise.  This  difference  is  plainly  shown  in  the 
following  experiment. 

A  disk  of  wood  or  metal  is  provided  with  two  circular  rows  of  pegs, 
equally  spaced  in  one  row  at  intervals  of  about  i  cm.,  and  an  equal 
number  unequally  spaced,  at  various  irregular  intervals,  in  the  other 
(Fig.  223).  The  disk  is  mounted  on  a  rotator  and  rapidly  whirled, 
while  the  edge  of  a  small  card  is  held  lightly  against  the  regular  row 
of  pegs,  then  against  the  other  row.  In  contact  with  the  regular  row 
the  card  is  forced  to  execute  regular  periodic  vibrations;  and  it  then 
gives  out  a  musical  sound  of  definite  pitch,  which  varies  with  the 
speed  of  the  disk.  This  musical  note  is  readily  distinguished  from 
the  rattling  noise  which  accompanies  it.  In  contact  with  the  irregular 


FIG.  223. 


324  SOUND 

row  the  vibration  of  the  card  is  also  irregular,  and  only  a  noise  is 
produced.  (A  disk  provided  with  two  rows  of  holes  instead  of  pegs 
serves  equally  well  for  the  experiment,  the  sound  being  produced 
by  directing  a  jet  of  air  against  the  holes.) 

A  musical  sound  is  one  that  continues  of  uniform  loud- 
ness,  pitch,  and  quality  for  an  appreciable  length  of  time, 
and  does  not  change  irregularly.  Uniform  loudness  or 
intensity  is  due  to  a  constant  amplitude  of  vibration  (Art. 
265);  uniform  pitch,  to  a  constant  rate  of  vibration;  and 
uniform  quality,  to  a  constant  mode  of  vibration,  i.e.  to 
vibrations  which  are  all  simple  or  all  of  the  same  complex 
character.  (We  shall  presently  learn  that  the  vibrations 
of  a  sounding  body  rarely  consist  of  a  simple  to-and-fro 
motion,  like  that  of  a  pendulum  bob.  They  are  generally 
very  complex  indeed,  and  this  complexity  is  reproduced 
in  the  vibrations  of  the  air  particles  in  the  sound  wave.) 
A  musical  sound  is  often  called  a  tone  or  note. 

A  noise  consists  of  a  confused  mixture  of  sounds,  due  to 
an  extremely  complex  and  rapidly  changing  vibration  of 
the  sounding  body.  Owing  to  the  unsteady  and  non- 
periodic  character  of  the  vibrations,  a  noise  has  no  definite 
pitch  or  wave  length. 

272.   Relative  and  Absolute  Pitch.  Musical  Intervals.  — 

The  pitch  of  a  note  may  be  expressed  either  relatively  or 
absolutely.  It  is  expressed  relatively  by  stating  its  rela- 
tion to  some  other  note,  generally  the  keynote  of  the  mu- 
sical composition  in  which  it  occurs.  The  correspondence 
between  relative  pitch  and  relative  rates  of  vibration  is 
easily  shown  by  means  of  the  disk  siren. 

This  is  a  disk  pierced  with  two  or  more  circular  rows  of  holes, 
equally  spaced  in  each  row.  It  is  customary  to  provide  four  rows, 
having  respectively  24,  30,  36,  and  48  holes.  While  the  disk  is 
revolved  at  constant  speed,  a  jet  of  air  is  directed  against  the  different 


PROPERTIES   OF   MUSICAL  SOUNDS 


325 


rows  of  holes  in  succession  (Fig.  224).  A  note  is  produced  by  the 
regularly  recurring  puffs  of  air  which  escape  through  the  holes,  each 
puff  producing  a  sound  wave.  The  notes  from  the  four  rows  bear 
to  one  another  the  pitch  relation  expressed  by  the  syllables  do,  mi,  sol, 
do',  the  interval  between  the  first  and  the  last  being  an  octave.  As 
the  speed  of  the  disk  is  increased  the  pitch  of  all  the  notes  rises,  but 
their  relation  to  one  another  remains  unaltered. 

Thus  we  learn  that,  if  the  interval 
between  two  notes  is  an  octave,  the 
vibration  rate  of  the  higher  is  always 
twice  that  of  the  lower,  and,  further, 
that  the  notes  do,  mi>  sol,  do'  are 
produced  by  vibrations  whose  ratios 
to  one  another  are  expressed  by  the 
numbers  24,  30,  36,  and  48,  or  by  the 
smaller  numbers  4,  5,  6,  8. 

The  absolute  pitch  of  a  note  is 
measured  by  the  number  of  vibra- 
tions of  the  sounding  body  per  sec- 
ond. This  is  called  the  vibration 
number  or  frequency  of  the  note.  To  find  the  frequency 
of  any  note  of  the  siren  we  have  only  to  multiply  the 
number  of  holes  in  the  row  by  the  number  of  revolutions 
of  the  disk  per  second.  The  vibration  number  of  a  fork 
can  be  determined  by  causing  the  fork  to  make  a  perma- 
nent record  of  its  vibrations,  as  in  the  accompanying  labo- 
ratory exercise.  The  frequency  of  the  C  fork  corresponding 
to  middle  C  of  the  piano  or  organ  is  256,  i.e.  the  prongs 
of  this  fork  swing  outward  and  inward  256  times  in  a 
second.  (The  middle  C  of  musical  instruments  is  slightly 
higher  than  this.) 

The  experiment  with  the  siren  teaches  that  the  relative 
pitch  of  any  two  notes  is  determined  by  the  ratio  of  their 
frequencies.  The  ratio  of  the  greater  frequency  to  the  less 


FIG.  224. 
Disk  Siren. 


326  SOUND 

is  called  the  interval  between  the  notes.  For  example, 
the  octave  interval  is  always  2  and  the  do-sol  interval  f 
or  f .  A  person  with  a  musically  trained  ear  recognizes 
relative  pitch  with  great  accuracy,  not  as  a  numerical 
ratio,  but  as  a  direct  mental  impression. 

273.  The  Major  Diatonic  Scale.  —  When  two  or  more 
notes  are  sounded  together  the  effect  upon  the  ear  may  be 
either  pleasant  or  disagreeable.  If  the  keys  of  a  piano 
are  struck  at  random,  the  effect  will  certainly  be  disagree- 
able. This  shows  that  the  pleasing  or  harmonious  combina- 
tions of  sounds  are  comparatively  few  in  number,  while 
the  discordant  combinations  are  limitless. 

The  harmonious  musical  intervals  can  all  be  expressed 
as  the  ratio  of  small  whole  numbers,  such  as  f ,  f ,  J,  etc. 
These  are  called  simple  ratios,  as  distinguished  from  the 
ratio  of  larger  numbers,  e.g.  -j-f .  The  most  perfect  harmony 
is  that  of  a  note  and  its  octave,  and  the  interval  in  this 
case  is  the  simplest  possible. 

Any  three  notes  whose  frequencies  are  in  the  ratio  4,  5, 
6   form  a   major   triad.     This   combination  is   especially 
pleasing,  and  is  the  basis  of  the  major  diatonic  scale,  which 
is  made  up  of  three  such  triads,  as  follows: 
do  mi  sol 

fa  la         do' 

sol  si  re' 

If  we  reduce  the  last  note,  re' ,  one  octave,  to  bring  it 
within  the  same  octave  as  the  others,  we  have  the  complete 
scale.  The  following  table  gives  first  the  relative  fre- 
quencies of  the  notes  in  terms  of  the  smallest  whole  num- 
bers in  which  they  can  all  be  expressed,  second  the  interval 
between  each  note  of  the  scale  and  the  first  or  keynote, 
third  the  interval  between  each  note  and  the  preceding 


PROPERTIES   OF   MUSICAL  SOUNDS  327 

one.     This  table  should  be  memorized,  as  a  knowledge  of 
it  is  presupposed  in  the  discussion  of  later  topics. 


do 

re 

mi 

fa 

«rf 

la 

si 

do' 

1.    24 

27 

30 

32 

36 

40 

45 

48 

2.     i 

9 

t 

f 

f 

5 
3 

¥ 

2 

3.    i 

t. 

¥ 

H 

t 

¥ 

f 

if 

It  will  be  observed  that  the  intervals  between  successive 
notes  are  of  three  different  magnitudes,  namely  f ,  -9°-,  and 
|f.  The  first  two  are  only  slightly  unequal  and  are  called 
whole  tones;  the  third  is  considerably  smaller  and  is  called 
a  semitone.  The  eighth  note  completes  the  octave,  and  is 
at  the  same  time  the  first  of  another  series  of  eight  notes, 
each  of  which  is  an  octave  above  the  note  of  the  same  name 
in  the  preceding  series.  The  scale  may  thus  be  repeated 
both  upward  and  downward,  through  as  many  octaves  as 
is  desired.  The  intervals  remain  the  same  whatever  the 
absolute  pitch  of  the  keynote  may  be.  When  middle  C 
(as  sounded  by  a  tuning  fork)  is  taken  as  the  keynote, 
the  letter  names  and  the  vibration  numbers  of  the  notes 
of  the  scale  are  as  given  below. 


Position  on  the  staff, 


Letter  names,  c'       d'        e'        f          g'        a!          V       c' 

Syllable  names,  do       re       mi       fa         sol       la          si       do' 

Vibration  numbers  256      288     320     341^     384     426!     480     512 
Vibration  ratios,  i         f         4         I  I ;         t          ¥        2 

Intervals  between 

successive  notes,  I       ¥       If  t       ¥  I       If 

274.  The  Equally  Tempered  Scale.  —  The  white  keys  of  a  piano 
or  an  organ  have  the  semitone  intervals  in  the  right  place  for  musical 
compositions  written  in  the  "key  of  C."  For  compositions  in  any 
other  key  one  or  both  of  these  intervals  would  be  out  of  place,  and  a 
whole  tone  interval  would  come  where  a  semitone  belongs.  It  is 


328  SOUND 

therefore  necessary  to  introduce  other  notes  within  the  octave. 
These  notes  are  called  sharps  or  flats,  and  are  played  by  means 
of  the  black  keys.  Five  notes  are  thus  added  in  the  octave,  forming 
the  chromatic  scale  — 

CC#DD#EFF8GG#AA#B      C 

It  is  further  necessary  to  change  the  intervals  slightly,  so  as  to 
make  them  all  exactly  equal;  otherwise  the  interval  V  would 
sometimes  occur  where  the  interval  §•  should  be,  and  vice  versa. 
The  scale  thus  modified  is  called  the  equally  tempered  scale,  to  dis- 
tinguish it  from  the  natural  scale  first  described.  The  tempered 
scale  is  a  compromise,  necessitated  by  the  practical  requirements  of 
musical  instruments.  In  the  physical  study  of  sound,  pitch  is  always 
referred  to  the  natural  scale. 

275.   Limits  of  Audibility.    Range  of  Pitch  Used  in  Music.  —  The 
human  ear  is  not  sensitive  to  all  rates  of  vibration.     If  the  rate  is 
below  a  certain  minimum,  either  no  sound  is  heard  or  only  separate 
pulsations.     This  minimum   varies   with 
different  persons,   but  generally  lies   be- 
tween  1 6  and  30  vibrations  per  second. 
'IG'  2WhiTtkalt°n'S          lt  is  an  interesting  fact  that  the  muscles 
vibrate  when  in  action,  sounding  a  note 

near  the  lower  limit  of  audibility.  This  note  can  be  distinctly  heard 
as  a  rapid  pulsation  by  pressing  the  palms  of  the  hands  firmly  over  the 
ears.  The  sound  comes  from  the  muscles  of  the  arms,  which  are  then 
contracted.  The  lowest  note  of  a  piano  (A  of  the  fourth  octave  below 
middle  C)  is  near  the  limit  of  audibility,  its  frequency  being  26.6. 

There  is  also  a  higher  limit  of  frequency  above  which  the  ear  is 
not  affected  and  no  sound  is  heard.  This  upper  limit  varies  much 
more  widely  with  different  persons  than  the  lower,  ranging,  as  experi- 
ment shows,  from  12,000  to  30,000  vibrations  per  second.  The 
shrill  cry  of  a  bat  and  the  high-pitched  noises  of  many  insects  are 
inaudible  to  some  whose  ears  are  normal  for  sounds  of  ordinary  pitch. 
The  upper  limit  of  audibility  is  determined  by  gradually  raising  the 
pitch  of  a  Gal  ton  whistle  (Fig.  225)  until  the  note  is  no  longer  heard. 
When  the  experiment  is  tried  before  a  class  it  will  always  be  found 
that  the  note  is  clearly  heard  by  some  after  it  has  become  inaudible 
to  others.  The  highest  note  of  a  piano  is  the  fourth  C  above  middle 


PROPERTIES   OF   MUSICAL   SOUNDS  329 

C,  and  its  vibration  number  is  256  X  24  =  4096.     Pitch  is  not  easily 
or  definitely  appreciated  beyond  this  limit. 

A  sound  of  any  pitch  will  be  inaudible  if  the  amplitude  of  vibra- 
tion is  too  small.  Lord  Rayleigh  found  that  the  faintest  audible 
sound  has  an  amplitude  of  vibration  of  the  air  particles  of  about  one 
millionth  of  a  millimeter.  "In  an  extremely  loud  sound,  such  as  that 
of  a  steam  whistle  heard  close  at  hand,  the  amplitude  of  vibration 
is  probably  less  than  i  mm."  —  H attack. 

276.  The  Relation  between  Pitch,  Wave  Length,  and 
Velocity.  —  Consider  what  happens  while  a  sounding  body 
vibrates  for  one  second.     If  n  denotes  the  frequency  of 
the  body,  a  train  of  n  waves  will  be  sent  out  during  the 
second.     At  the  end  of  the  second  the  last  of  these  waves 
is  on  the  point  of  leaving  the  body,  and  the  first,  having 
traveled  for  one  second,  has  gone  a  distance  equal  to  the 
velocity  of  sound  in  the  medium.     Let  v  denote  this  veloc- 
ity and  /  the  length  of  the  wave;  then,  since  the  n  waves 
extend  over  the  distance  v,  the  length  of  each  wave  is  one 

iih  of  v;  i.e.  I  =  ~,  or  v  =  In. 

It  follows  that  the  wave  length  in  a  given  medium  is 
inversely  proportional  to  the  frequency,  e.g.  raising  the  pitch 
of  a  sound  one  octave  reduces  its  wave  length  one  half. 
It  follows  also  that  the  wave  length  of  a  sound  of  given 
pitch  is  proportional  to  the  velocity  of  sound  in  the  medium. 
Thus  the  wave  length  of  middle  C  in  air  at  o°  is  1090  -f- 
256  =  4.26  ft. ;  in  water  it  is  about  4.3  times  as  great,  and 
in  steel  15  times  as  great,  or  64  ft. 

277.  Interference  of  Sound.  —  Under  ordinary  circum- 
stances,  when    many  bodies  are  vibrating  at    the   same 
time,  the  sound  from  each  is  heard  just  as  if  the  others  were 
silent.     The  leader  of  an  orchestra  recognizes  the  notes 
from  each  instrument,  though  all  are  sounding  together. 


330 


SOUND 


It  is  evident,  therefore,  that  the  same  body  of  air  may,  at 
a  given  instant,  be  taking  part  in  the  transmission  of  any 
number  of  sounds  in  any  and  all  directions,  regardless  of 
the  relative  lengths  and  intensities  of  the  different  waves. 
The  actual  motion  of  each  air  particle  is,  of  course,  the 
resultant  of  all  the  component  motions  that  it  would  have 
at  the  instant  in  transmitting  the  waves  individually. 

The  union  of  two  or  more  sets  of  sound  waves  often  pro- 
duces certain  special  effects,  which  differ  under  different 
conditions.  These  effects  include  harmony,  discord,  and 
quality,  the  causes  of  which  remain  to  be  considered,  and 
the  phenomena  of  interference,  beats,  and  resonance. 

The  simplest  phenomenon  of  this 
class  is  the  mutual  destruction  of 
two  trains  of  sound  waves  in  cer- 
tain regions,  producing  silence  in 
those  regions.  This  is  called  de- 
structive interference.  Any  tuning 
fork  affords  an  excellent  example. 
When  a  vibrating  fork  is  held 
close  before  the  ear  and  slowly  ro- 
tated about  the  stem  as  an  axis, 
the  sound  swells  to  a  maximum 
and  dies  away  to  silence  four  times 
during  one  complete  rotation.  With 
the  fork  held  in  a  position  of 

silence,  the  sound  is  restored  by  covering  either  prong  with  a  small 
paper  cylinder,  care  being  taken  not  to  touch  the  prongs,  as  this 
would  stop  the  vibration.  The  phenomenon  can  be  shown  before 
a  class  with  the  aid  of  a  resonance  jar,  tuned  in  unison  with  the  fork 
(Fig.  226).  These  curious  effects  are  explained  with  the  aid  of 
Figs.  227  and  228,  which  represent  the  waves  about  a  sounding  fork, 
the  prongs  pointing  toward  the  observer.  As  the  prongs  move  apart 
a  compression  is  set  up  on  the  outside  of  each,  and  a  rarefaction 
between  them;  as  they  move  toward  each  other  the  effects  are  inter- 
changed. The  fork  thus  sends  out  four  trains  of  waves  of  equal  wave 


FIG.  226.  —  Interference. 


PROPERTIES   OF  MUSICAL   SOUNDS  331 

length  and  intensity,  but  with  the  waves  of  adjacent  trains  in  opposite 
phase,  i.e.  the  waves  of  one  train  are  half  a  wave  length  in  advance  of 


FIG.  227.  —  Component  Waves     FIG.  228.  —  Resultant  Waves 
about  a  Tuning  Fork.  about  a  Tuning  Fork. 

the  waves  of  the  adjacent  train.  If  the  adjacent  trains  had  no  effect 
upon  each  other,  the  compressions  of  one  would  travel  outward  side 
by  side  with  the  rarefactions  of  the  other  and  vice  versa,  as  repre- 
sented in  Fig.  227.  But  in  the  region  where  they  meet,  their  opposing 
tendencies  constantly  neutralize  each  other;  for  the  compression 
would  be  transmitted  by  a  forward  motion  of  the  air  particles  and  the 
rarefaction  by  an  equal  backward  motion  at  the  same  time.  Hence 
the  air  in  this  region  remains  at  rest  and  there  is  neither  compression 
nor  rarefaction.  Silence  is  thus  the  result  of  the  interference  of  the 
waves  with  each  other. 

278.  Periodic  Interference  or  Beats.  —  A  more  impor- 
tant case  of  interference  is  that  of  two  trains  of  waves  of 
very  slightly  unequal  wave  length.  Two  forks  of  the  same 
pitch  will  serve  to  illustrate.  If  they  are  of  exactly  the 
same  pitch,  they  sound  together  as  one;  but  if  there  is  a 
very  slight  difference  (as  when  the  prongs  of  one  are  loaded 
with  a  bit  of  soft  wax),  their  united  sound  periodically 
swells  and  dies  away  in  strongly  marked  pulsations.  This 
is  explained  as  follows:  We  will  suppose  that  two  middle 
C  forks  are  used  and  that,  by  loading  with  wax,  the  fre- 
quency of  one  is  reduced  from  256  to  255.  At  intervals 


332  SOUND 

of  one  second,  the  forks  "keep  step"  in  their  vibrations; 
and  the  waves  that  they  then  set  up  approximately  coin- 
cide, compressions  with  compressions  and  rarefactions 
with  rarefactions,  as  at  A  and  C  (Fig.  229),  in  which  the 
compressions  are  represented  as  crests  and  the  rarefac- 
tions as  troughs.  These  waves  unite  in  resultant  waves 
of  increased  intensity,  as  represented  at  X  and  Z.  Half  a 
second  after  each  coincidence  the  forks  vibrate  oppositely, 


FIG  229.  — Interference  of  Two  Trains  of  Sound  Waves. 

the  compressions  produced  by  each  approximately  coin- 
ciding with  the  rarefactions  produced  by  the  other,  as  at 
B;  and  the  resultant  waves  are  of  diminished  intensity. 
A  complete  set  of  intensified  and  weakened  resultant  waves 
is  thus  sent  out  from  the  forks  during  each  second.  This 
constitutes  one  beat. 

Beats  may  therefore  be  defined  as  regularly  recurring 
pulsations  of  sound  caused  by  the  successive  reinforcement 
and  interference  of  two  trains  of  sound  waves  differing 
slightly  in  wave  length  or  pitch.  The  number  of  beats 
per  second  is  equal  to  the  difference  between  the  frequen- 
cies of  the  two  sounds.  Beats  are  produced  when  two 
wires  of  a  sonometer,  tuned  nearly  to.  unison,  are  sounded 
together.  They  become  less  frequent  and  finally  cease 
as  the  unison  is  made  more  nearly  perfect. 

279.  Beats  the  Cause  of  Discord.  —  When  beats  become 
too  frequent  to  be  separately  distinguished  by  the  ear,  the 


PROPERTIES   OF   MUSICAL   SOUNDS  333 

constituent  notes  blend  into  a  rough,  unpleasant  sound, 
or  discord.  This  can  be  shown  by  means 'of  a  sonometer 
(Fig.  230) .  Its  two  wires  are  first  tuned  to  unison,  then 
the  pitch  of  one  is  gradually  raised,  either  by  shortening 
the  vibrating  portion  with  a  movable  bridge  or  by  increas- 
ing the  tension.  The  beats  grow  more  rapid  as  the  inter- 
val between  the  notes  increases,  and  presently  merge  into 
a  discord.  As  the  interval  is  further  increased,  the  sound 


becomes  less  discordant,  then  harmonious,  then  again  dis- 
cordant, etc.  The  first  harmonious  interval  is  f  (mi-sol), 
the  second  f  (do-mi),  the  third  f  (do-fa),  etc.,  as  we  have 
already  learned  in  studying  the  musical  scale. 

That  discord  is  always  due  to  beats  was  shown  by  the 
investigations  of  the  noted  German  physicist,  von  Helm- 
holtz.  A  rapidly  pulsating  sound  is  disagreeable  to  the 
ear,  just  as  a  flickering  light  is  to  the  eye.  But  if  beats 
cease  and  the  sound  becomes  steady  when  the  interval 
between  two  notes  is  increased  to  f ,  how  is  the  recurrence 
of  discord  at  greater  intervals  to  be  accounted  for?  The 
answer  is  that  the  notes  of  all  musical  instruments  are 
complex.  The  principal  constituent  of  a  note  is  always 
accompanied  by  one  or  more  —  generally  several  —  higher 
constituents,  and  beats  are  possible  between  these  higher 
constituents  of  different  notes.  We  shall  have  occasion 
to  refer  to  this  again  when  we  take  up  the  study  of  the  com- 
plex character  of  sounds. 


334  SOUND 

280.  The  Frequency  of  Vibrating  Strings.  —  The  notes 
of  nearly  all  musical  instruments  are  produced  by  the  vibra- 
tion of  strings  (including  wires)  or  of  columns  of  air.  The 
laws  of  vibration  of  strings  are  therefore  of  special  impor- 
tance in  the  study  of  musical  sounds,  and  now  demand  our 
attention. 

Although  the  sound  of  a  stringed  instrument  comes  almost 
wholly  from  the  body  of  the  instrument,  or  from  some  part 
of  the  body,  the  pitch  of  the  notes  is  determined  by  the 
rate  of  vibration  of  the  strings.  It  is  well  known,  in  a 
general  way,  that  the  pitch  of  a  string  is  raised  by  shorten- 
ing the  part  that  vibrates  or  by  increasing  its  tension,  and 
that  a  light  string  sounds  a  higher  note  than  a  heavier  string 
of  the  same  length  and  under  the  same  tension.  The  effect 
of  length  is  illustrated  by  the  different  notes  obtained  from 
the  same  string  of  a  violin,  mandolin,  or  guitar,  by  vary- 
ing the  length  of  the  vibrating  portion  with  the  finger; 
the  effect  of  tension,  by  the  use  of  the  tightening  pegs  in 
tuning  the  strings ;  and  the  effect  of  mass,  by  the  different 
strings  of  the  instrument,  the  heaviest  giving  the  lowest 
note.  Definite  information  concerning  these  effects  is 
readily  obtained  with  the  aid  of  a  sonometer  (Fig.  230). 
The  effect  of  length  is  particularly  important  for  our  pur- 
pose and  must  be  carefully  noted. 

Effect  of  Length.  —  If  the  two  wires  of  a  sonometer  are 
tuned  to  unison  and  the  bridge  is  then  placed  under  one 
of  them  at  its  mid-point,  it  will  be  found  that  the  note  of 
this  wire  has  been  raised  exactly  an  octave.  (In  this 
and  similar  experiments  the  interval  may  be  judged  by  the 
ear,  the  two  notes  being  sounded  together  or  in  quick  suc- 
cession, or  the  pitch  of  each  note  can  be  determined  by  com- 
parison with  a  set  of  forks.)  If  the  note  sounded  by  the 
whole  length  of  the  wire  is  taken  as  the  first  note  of  the  scale, 


IjROPERTIES   OF   MUSICAL   SOUNDS  335 

or  do,  re  is  sounded  by  |  of  the  length,  mi  by  -f ,  fa  by  f , 
sol  by  |,  la  by  f ,  and  si  by  -f%.  These  length  ratios  are 
the  reciprocals  of  the  intervals  between  the  corresponding 
notes  (Art.  273).  The  relation  is  general:  The  tension  re- 
maining the  same,  the  frequency  of  a  string  or  wire  varies 
inversely  as  its  length. 

Effect  of  Tension.  —  If  the  sonometer  is  provided  with  some  means 
of  measuring  tension,  it  will  be  found  that  the  pitch  of  a  string  is 
raised  one  octave  by  increasing  its  tension  fourfold.  The  general 
law  is  that,  other  conditions  remaining  constant,  the  frequency  of  a  string 
varies  directly  as  the  square  root  of  Us  tension. 

Effect  of  Mass.  —  If  a  sonometer  is  strung  with  piano  wires  whose 
diameters  are  in  the  ratio  f,  and  the  lengths  and  tensions  of  the 
wires  are  the  same,  the  note  of  the  smaller  will  be  an  octave  above 
that  of  the  larger.  Since  the  cross-sections  of  the  wires  are  as  the 
squares  of  their  diameters,  their  masses  are  also  as  the  squares  of 
their  diameters.  Hence  the  masses  of  the  wires  are  in  the  ratio  f ; 
and  since  their  frequencies  are  in  the  ratio  i,  it  follows  that  their 
frequencies  are  inversely  proportional  to  the  square  roots  of  their 
masses.  In  general:  The  lengths  and  tensions  of  two  strings  being 
the  same,  their  frequencies  are  inversely  proportional  to  the  square  roots 
of  their  masses.  With  strings  of  the  same  material,  their  frequencies 
are  inversely  proportional  to  their  diameters. 

Since  the  laws  of  strings  are  only  special  cases  of  accelerated  motion, 
it  is  clear  that  they  must  be  in  accord  with  the  general  laws  of  dynam- 
ics. The  proof  of  this  involves  mathematical  work  beyond  the  scope 
of  elementary  physics.  It  will  be  an  instructive  exercise,  however, 
to  account  in  a  general  way  for  the  increase  of  frequency  with  decrease 
of  length,  increase  of  tension,  and  decrease  of  mass  per  unit  length. 
(This  is  left  as  an  exercise  for  the  pupil.) 

PROBLEMS 

1.  Why  is  the  sound  of  a  fork  restored  in  the  position  of  silence  when 
one  prong  is  covered?     Would  you  expect  to  find  new  positions  of  silence 
about  the  single  prong?     Text  your  conclusion. 

2.  Why  is  the  pitch  of  a  fork  lowered  by  loading  its  prongs  with  a  piece 
of  wax? 


336  SOUND 

3.  Compute    the  wave  lengths  of   the  following   notes  in  air  at  20°: 
Ci  (  =  32),  C  (  =  64),  cv  (=  4096),  and  the  highest  audible  sound,  assuming 
it  to  be  30,000. 

4.  A  string  i  m.  long  sounds  C  (=  64)  under  a  certain  tension.     Com- 
pute the  frequency  of  the  notes  sounded  by  f ,  f ,  i,  |,  |,  fc  and  |  of  its  length, 
the  tension  remaining  the  same.     All  these  notes  but  one  are  notes  of  the 
diatonic  scale  in  the  first  three  octaves  above  the  note  of  the  whole  wire. 
Calling  the  lowest  note  do,  what  are  the  syllable  names  of  the  others? 

5.  The  pitch  of  the  whistle  or  bell  of  a  passing  locomotive  or  gong  of  a 
trolley  car  drops  as  the  source  of  sound  passes  the  observer  and  changes 
from  approaching  to  receding.     This  phenomenon  is  called,  after  its  dis- 
coverer, the  "  Doppler  effect."     Explain  it. 

6.  When  sound  travels  from  colder  to  warmer  air,  does  its  wave  length 
change?     Does  its  frequency  change?     Explain. 

7.  Examine  the  wires  of  a  piano  and  find  in  what  respects  they  differ 
from  one  another.     What  means  are  employed  to  obtain  the  wide  range 
of  pitch  from  the  lowest  note  to  the  highest? 

281.  Fundamental  Tone  and  Overtones.  — When  a  string 
swings  to  and  fro  as  a  whole,  it  sounds  its  lowest  note  for 
the  given  length  and  tension.  This  is  called  its  funda- 
mental tone.  A  string  can  also  vibrate  in  two  or  more 
equal  segments;  and  the  notes  thus  produced  are  higher 
than  the  fundamental,  and  are  called  overtones  or  harmon- 
ics. This  mode  of  vibration  can  be  studied  to  advantage 
by  means  of  a  long  rubber  tube  or  a  coil  of  wire.  When 
such  a  tube  is  made  fast  at  one  end  and  stretched  in  either 
a  vertical  or  a  horizontal  position,  it  can  be  thrown  into 
vibration  as  a  whole  by  impulses  properly  timed  with  the 
hand  at  the  free  end  (Fig.  231).  This  is  the  motion  of  a 
string  when  sounding  its  fundamental  tone.  When  the 
rate  of  the  impulses  is  doubled,  the  tube  vibrates  in  two 
equal  segments  (Fig.  232).  This  is  the  motion  of  a  string 
when  sounding  its  first  overtone;  and,  by  the  law  of  lengths, 
the  pitch  of  this  tone  is  the  octave  above  the  fundamental. 
Similarly,  under  impulses  which  are  three  times  as  rapid 


PROPERTIES   OF   MUSICAL   SOUNDS 


337 


as  at  first,  the  tube  vibrates  in  thirds;  under  impulses  four 
times  as  rapid  it  vibrates  in  fourths,  etc.     In  all  cases  the 


FIG.  231. — Vibration  as  a  Whole. 


FIG.   232.  —  Vibration  in  Two  Segments. 

vibrating  segments  are  separated  by  points,  called  nodes, 
which  remain  approximately  at  rest;  and  adjacent  seg- 
ments are  constantly  in  opposite  phases  of  their  motion, 
i.e.  when  any  segment  is  moving  down,  the  next  one  on  either 
side  is  moving  up,  etc.  The  frequency  varies  inversely  as 
the  length  of  the  segments,  or  directly  as  the  number  of 
segments. 

The  wire  of  a  sonometer  can  easily  be  made  to  vibrate 
as  a  whole  or  in  any  number  of  equal  segments  up  to  eight 
or  ten.  When  it  is  bowed  or  plucked  near  one  end  and,  at 
the  same  time,  is  lightly  touched  at  its  mid-point  with  a 


FIG.   233.  — String  Sounding  its  Second  Overtone. 

feather  or  the  tip  of  the  finger,  it  vibrates  in  halves,  sound- 
ing the  octave  above  its  fundamental,  even  after  the  bow 


338  SOUND 

and  the  finger  are  removed  (the  finger  being  removed  an 
instant  after  the  bow).  When  it  is  touched  at  one  third 
its  length  from  the  end  that  is  bowed,  it  vibrates  in  thirds 
(Fig.  233);  when  touched  at  one  fourth  its  length,  it  vi- 
brates in  fourths,  etc.  In  every  case  the  entire  string 
vibrates.  Its  division  into  segments  can  be  shown  at  a 
distance  by  placing  upon  it  small  folded  pieces  of  paper. 
These  "  riders"  are  thrown  off  by  the  vibration  except  at 
or  very  near  the  nodes. 

The  overtones  are  numbered  in  order,  beginning  with 
the  lowest.  If  we  call  the  fundamental  tone  do\,  the  series 
of  possible  overtones,  up  to  and  including  the  seventh,  is 
as  follows: 


OVERTONES 

FUNDA- 

MENTAL 

ISt 

and 

3rd 

4th 

5th 

6th 

7th 

No.  of  segments  

I 

2 

3 

4 

.  5 

6 

7 

8 

Frequency    

n 

2n 

3n 

4n 

5n 

6n 

?n 

8n 

Relative  pitch    

dot 

do2 

soh 

dos 

mia 

soh 

— 

do* 

282.    Simple    and    Complex    Sounds.     Quality.  —  The 

question  remains  whether  a  string  can  vibrate  as  a  whole 
and  in  segments  at  the  same  time.  This  is  answered  by 
the  following  experiments.  Let  the  wire  of  a  sonometer 
be  bowed  or  plucked  at  about  one  fourth  its  length  from 
one  end.  It  sounds  its  fundamental,  and  is  therefore 
vibrating  as  a  whole.  If,  while  still  vibrating,  it  is  lightly 
touched  at  the  center  with  the  finger  or  a  feather,  the  fun- 
damental ceases  and  the  first  overtone  is  heard;  hence  it 
must  be  vibrating  in  halves.  The  touch  at  the  center 
destroys  the  vibration  as  a  whole,  but  does  not  interfere 
with  the  vibration  in  halves,  since  this  is  the  position  of 


PROPERTIES  OF   MUSICAL  SOUNDS 


339 


the  node.  The  overtone  is  present  with  the  fundamental; 
but,  being  relatively  weak  and  blending  perfectly  with  the 
lower  note,  it  can  be  recognized  only  by  fixing  the  atten- 
tion upon  it,  and  then  only  by  a  trained  ear.  If  the  wire 
is  very  lightly  touched  at  the  center  when  plucked  at  the 
quarter,  it  is  possible,  by  skilful  manipulation,  to  sound 
the  fundamental  and  the  overtone  with  nearly  equal  inten- 
sity, and  both  can  then  be  distinguished  with  little  diffi- 
culty. The  behavior  of  a  string  or  wire  when  sounding 


FIG.  234.  —  Complex  Vibration  of  a  String  as  a  Whole  and  in  Halves. 

its  fundamental  and  first  overtone  is  shown  by  the  full 
lines  of  Fig.  234.  The  vibration  as  a  whole  is  represented 
by  the  dotted  lines.  Upon  this  is  superposed  the  vibra- 
tion in  halves. 

When  the  wire  is  plucked  at  the  center,  then  touched  at 
that  point,  the  octave  is  not  heard.  The  reason  is  that 
vibration  in  halves  requires  a  node  at  the  center,  and  a 
node  can  not  exist  at  the  point  where  the  string  is  drawn 
aside.  When  the  wire  is  plucked  at  one  sixth  its  length 


340  SOUND 

from  one  end  and  touched  at  one  third  its  length,  the 
fundamental  is  quenched  and  the  second  overtone  is  heard. 

Higher  overtones  than  the  first  or  second  are  also  present  in  every 
case.  This  can  be  shown  as  follows.  Pluck  the  wire  near  one  end, 
and  immediately  touch  it  at  the  center.  The  first  overtone  is  heard. 
Again  pluck  it,  exactly  as  before,  and  touch  it  at  one  third  its  length. 
The  fundamental  and  the  first  overtone  are  quenched  by  the  touch, 
and  the  second  overtone  is  heard.  Repeat,  touching  the  wire  at  the 
quarter.  If  the  third  overtone  is  present  it  will  now  be  heard,  as  all 
lower  notes  are  quenched.  Similarly  the  presence  of  any  overtone 
can  be  determined  by  touching  the  vibrating  wire  at  the  point  where 
a  note  would  occur  for  that  overtone.  As  a  rule  the  first  six  or  eight 
overtones  can  be  detected  in  this  manner;  but  their  relative  intensity 
varies  greatly  with  the  place  where  the  wire  is  struck,  plucked,  or 
bowed,  and  also  with  the  character  of  the  impulse.  When  it  is  struck 
with  a  soft  mallet  or  plucked  with  the  finger  near  the  center,  the 
fundamental  is  loud  and  all  the  overtones  weak;  when  it  is  struck 
or  plucked  near  the  end  the  higher  overtones  are  much  stronger,  and 
especially  when  it  is  struck  with  a  hard  body  or  plucked  with  the 
finger-nail. 

In  listening  to  the  note  of  the  wire  in  the  above  experi- 
ments, the  pupil  has  doubtless  observed  that  it  sounds 
differently  when  produced  in  different  ways.  It  may  be 
equally  loud  in  the  different  trials,  and  it  does  not  vary 
in  pitch,  for  the  pitch  is  always  that  of  the  fundamental. 
The  difference  is  a  difference  of  quality  (Art.  271).  When 
the  wire  is  struck  with  a  rubber  mallet  or  plucked  near  its 
center,  the  tone  is  soft  and  mellow;  when  struck  or  plucked 
near  one  end,  the  tone  is  described  as  full,  rich,  or  bright; 
when  struck  with  a  hard  body  or  plucked  with  the  finger- 
nail close  to  one  end,  the  tone  loses  its  musical  character 
and  becomes  a  sharp,  discordant  jangle.  These  various 
qualities  are  plainly  due  to  the  overtones.  The  first  five 
overtones  are  in  harmony  with  one  another,  as  well  as  with 
the  fundamental  (see  table).  When  the  wire  is  sounding 


PROPERTIES   OF   MUSICAL   SOUNDS  341 

only  these,  the  effect  is  pleasing.  The  first  discordant 
overtone  is  the  sixth  of  the  series,  which  forms  a  discord 
with  both  the  fifth  and  the  seventh.  The  seventh  and 
ninth  overtones  are  in  harmony  with  each  other  and  with 
all  below  the  sixth;  the  eighth  and  the  tenth  are  discordant. 
As  we  go  higher  in  the  series,  the  discordant  overtones  are 
found  in  increasing  number. 

Sounds  are  in  general  complex,  and  their  quality  is  deter- 
mined by  the  pitch  and  relative  intensity  of  the  waves  of 
higher  frequency  which  accompany  the  fundamental,  A 
sound  without  overtones  is  called  a  simple  or  pure  tone. 
The  only  perfect  example  is  the  note  of  a  tuning  fork. 
The  overtones  of  a  fork  are  very  high.  They  sound  as 
a  clear,  shrill  note  when  a  fork  is  struck;  but  they  quickly 
die  away,  leaving  only  the  fundamental.  The  tones  of 
the  diapason  or  stopped  pipes  of  an  organ  are  the  nearest 
approach  to  simple  tones  used  in  music.  Their  overtones 
are  few  and  weak  and  affect  the  quality  but  little.  In 
general,  musical  sounds  are  rich  in  overtones,  or  harmonics, 
as  they  are  usually  called;  but  the  particular  combination 
of  overtones  and  their  relative  intensities  vary  with  differ- 
ent instruments. 

"  In  the  sound  of  a  violin  the  upper  harmonics  are  loud  and  piercing; 
the  nearer  harmonics  are  feeble,  and  the  fundamental  tone  stands 
apparently  alone,  but  rendered  penetrating  in  quality  by  the  high 
mass  of  harmonics.  In  a  piano  string  struck  by  an  elastic  soft  ham- 
mer the  harmonics  up  to  the  sixth  are  present;  the  seventh  is  obliter- 
ated, or  nearly  so,  by  the  hammer  being  made  to  strike  the  string  at 
a  spot  one  seventh  of  its  length  from  the  end  of  the  string;  and  the 
components  beyond  the  seventh  are  feebly  represented."  —  Daniell. 

283.  Harmony  Destroyed  by  Discordant  Overtones.  —  Two  notes 
sounded  together  are  discordant  when  beats  are  produced  by  the 
fundamentals  of  the  notes,  by  either  fundamental  and  an  overtone 
of  the  other,  or  by  an  overtone  of  each  (Art.  279).  It  can  be  shown 


342  SOUND 

that  these  numerous  possibilities  of  discord  limit  the  harmonious 
combinations  to  notes  whose  relative  frequencies  are  expressed  by 
simple  ratios  (Art.  273). 

284.  Vibrations  of  Bells.  —  A  body  sounds  its  fundamental  tone 
when  vibrating  in  the  least  number  of  segments  possible.     With 

strings   this    is   one    segment;  but    the    number 
varies  with  different  bodies.1    With  bells  and  bell- 
shaped  bodies,  such  as  glass  tumblers  and  bowls, 
it  is  four.     The  motion  of  the  rim  of  a  bell  is 
shown  in  Fig.  235,  the  point  where  the  clapper 
strikes  being  the  middle  of  a  segment.     The  over- 
tones of  bells  are  not,  as  a  rule,  multiples  of  the 
FIG.    235.  —  Seg-    fundamental,  and  their  musical  quality  is  conse- 
S^undin^  Bit!    <luently  imperfect.    This  is  particularly  noticeable 
Fundamental.     ,  when  bells  are  sounded  together.     The  discord 
between  their  overtones  is  very  marked,  although 
their  fundamental  tones  may  be  in  harmony.     On  this  account  the 
bells  of  a  chime  are  struck  singly.     The  familiar  pulsating  sound  of 
large  bells  is  due  to  the  interference  of  waves  from  different  seg- 
ments whose  frequencies  are  slightly  unequal,  on  account  of  slight 
irregularities  in  thickness  or  structure. 

III.   SYMPATHETIC  AND  FORCED  VIBRATIONS 
RESONANCE 

285.  A   Mechanical   Illustration.  —  Certain   mechanical 
actions  which  occur  with  sounding  bodies  can  be  shown 
on  a  visible  scale  with  two  pairs  of  pendulums  (Fig.  236), 
suspended  from  a  light  rod,  CD,  the  pendulums  of  each  pair 
being  of  equal  length.     The  rod  hangs   by  short  cords 
from  a  fixed  support. 

When  any  one  of  the  pendulums  is  set  in  vibration,  it 
pulls  the  rod  back  and  forth  in  unison  with  its  own  motion 
and  the  vibrating  rod  imparts  a  series  of  impulses  to  the 
other  pendulums.  The  effect  of  single  impulses  is  very 
slight;  but  their  periodic  repetition  causes  the  other  pendu- 
lum of  the  same  length  to  vibrate  with  steadily  increasing 


T 


I 


SYMPATHETIC  AND   FORCED  VIBRATIONS      343 

amplitude.  Meanwhile  the  amplitude  of  the  first  pendu- 
lum steadily  decreases,  as  it  imparts  its  energy  to  the  other. 
Since  the  two  pendulums  ^^^ 
vibrate  at  the  same  rate,  c  \ 
the  impulses  transmitted 
from  the  one  are  rightly 
timed  to  produce  a  cu- 
mulative effect  upon  the 
other.  The  latter  is  said 
to  vibrate  sympathetically, 
which  means  that  its  nat- 
ural rate  is  in  agreement 

with     the     impulses     tO    FlG-  236.  —  Mechanical  Illustration  of  Sym- 
...      .  pathetic  and  Forced  Vibrations. 

which  it  responds. 

The  other  two  pendulums  make  a  few  swings  with  in- 
creasing amplitude,  then  an  equal  number  with  decreas- 
ing amplitude,  coming  to  rest  again.  This  behavior  is 
repeated  indefinitely.  The  reason  is  that  the  impulses 
are  not  timed  in  agreement  with  the  rate  of  these  pendu- 
lums. A  few  successive  impulses  produce  a  cumulative 
effect;  but  these  are  followed  by  an  equal  number  which 
are  opposed  to  the  motion  already  produced  and  hence 
destroy  it.  Since  these  pendulums  can  not  be  forced  to 
vibrate  in  unison  with  the  impulses,  they  can  not  accumu- 
late any  considerable  store  of  energy. 

When  the  rod  is  struck  or  drawn  to  one  side  and  released, 
the  pendulums  being  at  rest,  it  vibrates  much  more  rapidly 
than  it  does  when  under  the  control  of  a  vibrating  pendu- 
lum. Thus,  although  the  rod  has  a  natural  rate  of  vibra- 
tion, it  does  not  persist  in  it  as  the  pendulums  do,  but 
yields  to  the  impulses  of  either  a  long  or  a  short  pendulum. 
The  motion  of  the  rod  when  thus  controlled  is  called  a 
forced  vibration. 


344  SOUND 

286.   Forced    Vibrations    of    Sounding    Bodies.  —  The 

sound  that  comes  from  a  table,  an  empty  box,  or  a  large 
board,  when  touched  by  a  vibrating  fork  is  due  to  the 
forced  vibration  of  the  wood.  Since  the  sound  thus 
produced  always  has  the  same  pitch  as  the  fork,  whatever 
this  may  be,  it  is  evident  that  wood,  especially  in  the  form 
of  a  thin  board,  is  easily  forced  to  vibrate  in  unison  with 
periodic  impulses  of  any  frequency,  -  -  a  behavior  like 
that  of  the  rod  in  the  above  experiment. 

The  music  of  stringed  instruments  is  due  to  forced  vibra- 
tions. "All  stringed  instruments,  with  few  exceptions, 
have  the  strings  stretched  between  pegs  which  are  fastened 
to  a  wooden  board  or  box.  Owing  to  the  vibrations  of  the 
strings,  and  the  resulting  motion  of  the  pegs,  this  board 
is  made  to  vibrate;  and,  since  these  vibrations  are  'forced,' 
they  imitate  more  or  less  closely  in  character  those  of  the 
strings.  But,  of  course,  there  are  differences  depending 
upon  the  thickness,  area,  stiffness,  etc.,  of  the  boards. 
Similarly,  if  there  is  a  box  or  cavity,  the  inclosed  air 
may  be  set  vibrating.  The  vibrations  of  this  'sounding' 
board  or  box  affect  the  surrounding  air  much  more  than  do 
those  of  the  fine  strings;  and  so  the  sound  we  hear  depends 
to  a  great  extent  upon  the  former  vibrations.  We  see, 
therefore,  the  reasons  why  the  violins  of  certain  makers 
have  such  great  value,  owing  to  their  skill  in  the  con- 
struction of  the  wooden  parts."  —  Ames. 

In  the  phonograph  we  have  a  twofold  application  of  forced* vibra- 
tions, first  in  making  the  original  "record,"  second  in  reproducing 
the  sound  from  a  copy  of  this  record.  In  making  the  record  the  sound 
waves  are  concentrated  upon  a  flexible  diaphragm,  to  which  a  cutting 
point  or  chisel  is  attached.  The  highly  complex  vibrations  of  the  air 
are  impressed  upon  the  diaphragm,  and  the  chisel  engraves  them  in 
wax  upon  a  revolving  disk  or  cylinder.  As  the  chisel  moves  forward 
with  the  diaphragm  it  makes  a  deeper  cut  in  a  spiral  groove;  as  it 


SYMPATHETIC   AND   FORCED   VIBRATIONS       345 

recedes  it  cuts  less  deeply.  The  records  made  for  use  are  reproductions 
of  the  original  in  some  hard  and  durable  material.  In  playing  a  piece 
of  music  the  record  revolves  under  the  point  of  a  blunt  needle,  which, 
as  it  travels  along  the  groove,  is  forced  to  repeat  the  vibrations  of 
the  recording  instrument.  The  vibrations  of  the  needle  are  trans- 
mitted by  a  lever  mechanism  to  the  thin  mica  diaphragm  of  the 
"reproducer"  or  sound  box,  and  by  the  diaphragm  to  the  air.  If 
the  phonograph  were  not  so  common,  we  should  marvel  at  the  exact- 
ness with  which  it  imitates  all  musical  instruments  either  singly  or 
together.  It  is  difficult  to  imagine  the  intricate  character  of  the 
vibrations  of  a  small  disk  in  reproducing  the  music  of  an  orchestra. 

287.  Sympathetic  Vibrations  of  Tuning  Forks  and  Strings.  —  A 
tuning  fork  may  be  made  to  vibrate  sympathetically  as  follows :  The 
stems  of  a  sounding  and  a  silent  fork  of  exactly  the  same  pitch  are 
touched  to  the  top  of  a  table  a  short  distance  apart.  After  one  or 
two  seconds  the  fork  that  was  sounded  is  stopped  with  the  fingers 
or  removed  from  the  table,  and  a  sound  is  then  heard  from  the  other 
fork.  If  the  sound  is  too  faint  to  be  heard  at  a  distance,  the  vibration 
of  the  fork  can  be  proved  by  touching  one  of  its  prongs  with  a  sus- 
pended pith  ball.  The  impulses  are  transmitted  through  the  vibrat- 
ing table  from  the  stem  of  one  fork  to  that  of  the  other.  Probably 
from  500  to  looo  such  impulses  are  required  to  produce  the  observed 
effect.  The  silent  fork  also  responds  to  the  other  when  they  are  held 
in  the  hands,  facing  each  other  but  not  touching.  In  this  case  the 
impulses  are  imparted  by  the  sound  waves  in  the  air.  These  experi- 
ments fail  with  forks  that  are  not  in  perfect  unison,  as  will  be  found 
by  trial  when  the  prongs  of  one  are  loaded  with  a  little  wax  at  the 
free  end. 

If  the  two  wires  of  a  sonometer  are  tuned  to  exact  unison,  either 
will  vibrate  sympathetically  when  the  other  is  plucked,  the  impulses 
being  transmitted  principally  through  the  body  of  the  instrument. 
As  the  pitch  of  one  of  the  wires  is  slowly  changed,  the  response  of  the 
silent  wire  immediately  becomes  very  faint  and  quickly  ceases. 

These  experiments  show  that  tuning  forks  and  strings,  like  the 
pendulums,  persist  in  meir  natural  period  of  vibration,  and  respond 
only  to  impulses  whose  frequency  is  the  same  as  their  own.  It  is 
owing  to  this  property  of  strings  that  violins,  pianos,  and  other 
stringed  instruments  are  possible. 


346 


SOUND 


288.  Vibrating  Columns  of  Air.  Resonance.  —  We  have 
seen  that  faint  sounds  are  greatly  intensified  by  the  forced 
vibration  of  thin  boards,  the  bodies  of  stringed  instru- 
ments, etc.  Sounds  are  also  reenforced  by  the  sympa- 
thetic vibration  of  partially 
inclosed  bodies  of  air,  as  in 
all  wind  instruments.  Such 
reinforcement  is  called  reso- 
nance, and  the  hollow  body 
in  which  it  occurs  is  some- 
times called  a  resonator. 
The  sounding  box  of  a  tun- 
ing fork  (Fig.  237)  is  a 
FIG.  237.  —  Set  of  Resonance  Boxes  and  resonator  of  such  dimensions 

Forks  of  Corresponding  Pitch.  . 

that   the   natural  period  of 

the  air  within  it  is  the  same  as  that  of  the  fork.  When 
the  vibrating  fork  is  held  before  the  open  end  of  the 
box,  the  reinforcement  of  the  sound  is  due  to  the  direct 
action  of  the  fork  upon  the  air. 

When  a  sounding  fork  is  held  at  an  end  of  a  large  glass 
tube  and  a  close-fitting  piston  is  steadily  moved  in  either 
direction  past  a  certain  point  (Fig.  238),  the  sound  from  the 
tube  swells  to  a  maximum  and  dies  away.  The  loudest 
response  occurs  when  the  length  of  the  column  is  such 
that  its  natural  period  is  the  same  as  the  period  of  the  fork. 


FIG.  238.  —  Adjustable  Resonance  Tube. 


The  fainter  sound  from  the  tube  when  the  column  is  a  little 
too  long  or  too  short  for  maximum  resonance  is  still  of  the 
same  pitch,  showing  that  a  body  of  air  can  be  forced  to 


SYMPATHETIC  AND   FORCED   VIBRATIONS       347 

depart  slightly  from  its  natural  period.  With  a  fork  of 
higher  pitch  the  length  of  the  resonating  column  of  air  is 
less,  with  a  fork  of  lower  pitch  it  is  greater.  It  will  thus 
be  found  that  the  frequency  of  an  air  column,  like  that  of 
a  string,  varies  inversely  as  its  length. 

The  vibration  of  air  columns  is  longitudinal,  and  may  be  regarded 
as  the  resultant  effect  of  a  train  of  waves  traveling  down  the  tube  and 
a  train  of  reflected  waves  traveling  back.  As  the  nearer  prong  of  the 
fork  advances,  it  sends  a  compression  down  the  tube.  The  front  of 
this  compression  must  travel  the  length  of  the  tube  and  back  while 
the  prong  is  advancing,  in  order  to  be  out  of  the  way  of  the  rarefac- 
tion, the  front  of  which  travels  down  the  tube  and  back  while  the 
prong  is  retreating.  Since  the  front  of  either  the  compression  or  the 
rarefaction  travels  twice  the  length  of  the  tube  during  a  half  vibration 
of  the  fork,  it  travels  four  times  that  length  during  a  complete  vibra- 
tion. But  the  distance  traveled  in  that  time  is  one  wave  length; 
hence  the  length  of  the  vibrating  air  column  is  one  fourth  of  a  wave  length. 
The  vibration,  however,  extends  beyond  the  open  end  of  the  tube  for 
a  short  distance,  equal  approximately  to  .3  the  diameter  of  the  tube; 
and  this  amount  is  added  as  a  "correction  for  the  diameter"  to  the 
measured  length  of  the  resonance  chamber. 

When  the  air  column  is  increased  to  three  fourths  of  a  wave  length, 
it  again  responds  to  the  sound  of  the  fork.  This  is  called  second 
resonance.  The  distance  from  the  mouth  of  the  tube  to  the  closed 
end  and  back  is  now  one  and  one  half  wave  lengths,  or  one  wave  length 
greater  than  for  first  resonance;  and  while  the  waves  are  traveling 
this  added  distance,  the  fork  makes  one  complete  vibration.  This 
brings  the  vibrations  of  the  fork  and  the  air  column  into  step  again; 
hence  the  resonance.  Similarly  there  will  again  be  resonance  if  the 
column  is  lengthened  to  five  fourths  of  a  wave  length,  or  indeed  to 
any  odd  number  of  quarter  wave  lengths. 

The  mode  of  vibration  of  the  air  is  shown  in  Fig.  239,  which  repre- 
sents the  condition  of  the  air  at  successive  quarter-period  intervals. 
AB  is  the  length  for  first  resonance,  AD  for  second  resonance.  AB, 
BC,  and  CD  are  each  one  fourth  of  a  wave  length.  At  B  and  D  there 
is  no  motion  and  the  change  of  density  is  the  greatest.  These  positions 
are  called  nodes.  (Compare  with  the  nodes  of  a  string  vibrating  in 


348 


SOUND 


segments,  as  shown  in  V.)  At  A  and  C  there  is  little  or  no  change  of 
density  and  the  amplitude  of  vibration  is  greatest.  These  positions 
are  called  antinodes.  The  arrows  in  II  and  IV  indicate  the  direction 
of  motion  of  the  air  particles.  At  the  instants  represented  in  I  and  III 
the  air  is  at  rest  throughout  the  tube.  BD  is  a  vibrating  segment  of 
the  air  column,  and  AB  is  a  half-segment.  The  compressions  and 
rarefactions  appear  and  disappear  in  succession  at  the  nodes.  They 


III 


. ::!,:•::..      ..iiiiHH!l!|iiililn|| 


Ilillllll 


V    A 


FIG.  239.  —  Four  Phases  of  a  Stationary  Wave. 

do  not  travel  as  they  do  in  waves.     For  this  reason  they  are  sometimes 
called  stationary  waves. 

A  tube  open  at  both  ends  also  acts  as  a  resonator  if  of  the  proper 
length.  The  length  can  be  adjusted  by  using  a  tube  made  in  two 
sections,  one  of  which  slips  inside  the  other.  A  roll  of  writing  paper 
will  serve  as  one  of  the  sections.  It  will  be  found  that  an  open 
resonance  tube  is  twice  as  long  as  a  closed  tube  responding  to  the  same 
note;  hence  the  length  of  an  open  tube  is  one  half  the  wave  length  of  its 
fundamental.  An  open  tube  has  an  antinode  at  both  ends,  and  when 
sounding  its  fundamental  it  vibrates  in  two  half -segments  with  a  node 


SYMPATHETIC  AND   FORCED   VIBRATIONS       349 

at  the  middle.  This  is  represented  in  the  figure  by  the  length  AC, 
assuming  an  open  end  of  the  tube  at  C.  The  length  for  second  reso- 
nance is  two  half-wave  lengths,  for  third  resonance  three  half-wave 
lengths,  etc.  A  resonance  tube,  either  open  or  closed,  sounds  its 
fundamental  when  adjusted  for  first  resonance,  its  first  overtone  when 
adjusted  for  second  resonance,  its  second  overtone  when  adjusted 
for  third  resonance,  etc. 

289.  Resonance  in  Response  to  Noises.  —  When  a  tube 
or  other  hollow  body,  as  a  vase,  tumbler,  or  sea  shell,  is 
held  to  the  ear,  a  faint  musical  sound  is  heard.  This  sound 
becomes  louder  when  noises  occur  in  the  vicinity,  but  its 
pitch  remains  constant.  Tubes  of  all  sizes  respond  to  the 
same  noise,  as  may  be  shown  by  holding  one  at  each  ear 
while  the  foot  is  scraped  on  the  floor;  but  the  note  from 
each  varies  with  the  size  of  the  tube,  and  is  its  fundamental 
tone.  Resonance  in  such  cases  is  the  response  of  the  air 
column  to  those  impulses  from  the  noise  which  agree  with 
its  own  period.  The  air  selects  a  series  of  impulses  from 
the  miscellaneous  assortment  present  in  the  noise,  and 
molds  them  into  a  musical  tone. 

Under  certain  conditions  the  vibrating  air  column  reacts 
upon  the  source  of  the  noise,  forcing  the  latter  into  regu- 
lar vibration  of  the  same  frequency  with  itself.  The  result 
is  a  clear,  strong,  musical  note,  familiar  to  all  in  the  music 
of  wind  instruments.  One  of  the  simplest  examples  of 
this  action  is  presented  by  blowing  across  the  end  of  a  nar- 
row tube.  As  long  as  the  stream  of  air  flutters  irregularly, 
the  note  from  the  tube  is  faint  and  the  rustling  noise  of 
the  air  current  is  comparatively  loud;  but  when  the  jet 
issues  from  the  lips  with  a  certain  velocity  and  strikes 
the  farther  edge  of  the  tube  at  the  proper  angle,  it 
yields  to  the  control  of  the  air  column  and  vibrates  regu- 
larly with  it.  The  note  from  the  tube  is  then  clear  and 


350 


SOUND 


strong  and  the  noise  is  nearly  or  quite  inaudible.  If  the 
tube  is  rather  long  it  can  be  made  to  sound  its  first  overtone 

IL_ by  harder   blowing,  and  perhaps    the 

second  by  still   harder.       A    stronger 
FIG.  240.  current,    like   a   string    under    greater 

tension,  requires  a  greater  force  to  deflect  it;  and,  when 
deflected,  it  vibrates  more  quickly.  The  mouthpiece  of 
a  whistle  serves  the  purpose  of  shaping  the  air  current 
and  directing  it  against  the  edge  of  the  opening  (Fig.  240). 
The  pitch  of  the  sound  is  determined  solely  by  the  dimen- 
sions of  the  air  chamber. 

Another  instructive  example  of  this  action  can  be  shown  with  a 
large  tin  tube,  several  feet  in  length,  and  a  rose  burner  (Fig.  241). 
The  numerous  small  flames  of  the  burner  flicker  rapidly 
and  unsteadily,  producing  a  buzzing  noise.  When  the 
tube  is  lowered  over  the  burner,  the  air  column  within 
it  responds  to  the  impulses  of  the  flames,  and  the  flames 
are  forced  to  vibrate  in  unison  with  the  column.  The 
tube  emits  a  loud,  deep  roar,  resembling  the  lower 
notes  of  a  pipe  organ,  and  the  vibrations  are  so  intense 
that  the  flames  are  sometimes  extinguished. 


290.  The  Pipe  Organ.  —  A  pipe  organ  is  in 
effect  an  assemblage  of  wind  instruments,  any 
number  or  combination  of  which  can  be  played 
at  the  same  time  by  one  person.  Each  of  these 
instruments  consists  of  a  set  of  pipes,  with  a 
pipe  for  each  note  of  the  scale,  and  the  entire 
set  is  brought  under  the  control  of  the  player 
by  drawing  a  "stop."  The  pipes  of  a  large 
organ  number  from  1500  to  2000  or  more.  The 
visible  pipes  at  the  front  are  intended  primarily  for  orna- 
ment, and  many  of  them  are  silent,  not  being  of  the  form 
and  dimensions  required  for  music.  Back  of  these,  arranged 


~  Ra°nsjj 
Resonance 


SYMPATHETIC  AND   FORCED   VIBRATIONS       351 

in  many  ranks,  are  the  "speaking"  pipes,  which  vary  in 
length  from  16  feet  down  to  an  inch  or  two,  according 
to  their  pitch. 

The  pipes  of  each  set  are  of  similar  form  and  construc- 
tion, and  their  notes  have  the  same  quality.  Some  sets  are 
made  of  wood  (Fig.  242),  others  of  metal  (Fig.  243);  some 
are  open  at  the  top,  oth- 
ers closed.  The  former 
are  called  open  pipes, 
the  latter  stopped  pipes. 
The  metal  pipes  of  one 
set  are  cylindrical ;  those 
of  another,  conical  or 
flaring  at  the  top,  like  a 
horn.  In  some  sets  the 
sound  is  produced  by  a 
vibrating  sheet  of  air,  as 
in  a  whistle,  in  others  by 
a  vibrating  reed,  as  in 
the  clarinet,  or  by  a 
double  reed,  as  in  the 
oboe.  These  and  many  Fia  242'  FlG"  243' 

other  special  features  of  construction  affect  the  number, 
pitch,  and  intensity  of  the  overtones,  and  hence  deter- 
mine the  quality.  The  organ  is  thus  made  to  imitate 
the  tones  of  all  other  wind  instruments,  and  (in  the 
tones  of  the  "vox  humana"  pipes)  even  the  sound  of  the 
human  voice. 

291.   Fundamental  Tones  and  Overtones  of  Organ  Pipes. 

—  The  air  in  the  resonance  chamber  of  an  organ  pipe 
always  vibrates  with  a  node  at  a  closed  end  and  an  anti- 
node  at  an  open  end,  as  in  other  tubes  (Art.  288).  There 


352 


SOUND 


is  always  an  antinode  at  the  lower  end,  on  account  of  the 
opening  at  that  end. 

In  an  open  pipe  sounding  its  fundamental  the  air  column 
vibrates  in  two  half-segments  with  a  node  at  the  middle 
(Fig.  2440).  The  wave  length  of  the  fundamental  is  there- 
fore twice  the  length  of  the  pipe  (corrected  for  the  diam- 


FIG.  244.  —  The  Vibrating  Segments  of  the  Air  Column  are 
Represented  by  the  Corresponding  Segments  of  a  String. 


eter).  An  open  pipe  sounding  middle  C  is  approximately 
two  feet  long.  The  octave  above  is  sounded  by  a  pipe  of 
half  that  length,  or  one  foot,  the  octave  below  by  a  four- 
foot  pipe,  etc.,  the  vibration  number  being  inversely  pro- 
portional to  the  length. 

In  a  closed  pipe  sounding  its  fundamental  the  air  column 
vibrates  in  one  half-segment  with  a  node  at  the  closed 


SYMPATHETIC   AND   FORCED   VIBRATIONS       353 

end  (Fig.  244^).  Hence  the  wave  length  is  four  times  the 
length  of  the  pipe.  A  closed  pipe  sounding  middle  C  is 
one  foot  long,  or  half  the  length  of  an  open  pipe  of  the  same 
pitch.  It  follows  that  the  pitch  of  a  closed  pipe  is  an  octave 
lower  than  that  of  an  open  pipe  of  equal  length. 

The  first  overtone  of  an  open  pipe  is  produced  when  the  air  column 
vibrates  with  one  whole  segment  between  the  half-segments  at  the 
ends,  as  in  b  of  the  figure.  For  the  second  overtone  there  are  two 
segments  between  the  half -segments,  as  in  c\  for  the  third  overtone 
three  segments;  etc.  Since  the  lengths  of  the  segments  of  the  funda- 
mental and  the  series  of  overtones  are  in  the  ratio  of  the  numbers  i, 
I,  i,  ?,  etc.,  the  frequencies  of  the  tones  are  in  the  ratio  of  i,  2,  3,  4, 
etc.  Hence  the  series  of  possible  overtones  with  open  pipes  is  the 
same  as  with  strings. 

In  a  closed  pipe  the  air  column  vibrates  in  one  and  one  half  seg- 
ments for  the  first  overtone  (Fig.  244,  e),  in  two  and  one  half  segments 
for  the  second  overtone,  in  three  and  one  half  segments  for  the  third, 
etc.  Since  the  lengths  of  the  segments  for  this  series  of  tones  are  in 
the  ratio  of  i,  i,  i,  etc.,  the  frequencies  are  in  the  ratio  of  i,  3,  5,  etc. 
Hence  with  closed  pipes  only  alternate  overtones  of  the  complete 
series  are  possible.  If  we  call  the  fundamental  doi,  the  overtones 
of  open  and  closed  pipes  are  as  follows: 

ist  2nd  3rd  4th  5th  etc. 
Open  pipe  doi  do2  sol*  do3  miz  sol3  etc. 
Closed  pipe  do\  —  sol2  —  miz  —  etc. 

With  so  many  overtones  missing,  the  fundamental  of  a  closed  pipe  is 
very  prominent  and  the  quality  is  deep  or  grave. 

Most  of  the  facts  stated  above  can  be  illustrated  by  means  of  an 
experimental  pipe  and  a  small  bellows.  The  pipe  should  be  provided 
with  a  piston  for  varying  the  length  of  the  air  column.  With  a  light 
air  pressure,  only  the  fundamental  of  the  pipe  is  heard;  with  a  greater 
pressure,  this  gives  place  to  the  first  overtone;  and,  with  still  greater 
pressure,  the  second  overtone  is  sounded. 

292.  Other  Wind  Instruments.  —  Most  wind  instruments  are 
provided  with  but  one  tube,  the  different  notes  being  produced  either 
by  varying  the  length  of  the  tube  or  by  sounding  overtones. 


354 


SOUND 


In  the  trombone  (Fig.  245)  one  part  of  the  tube,  L,  slides  within 
the  other,  S,  and  the  tube  is  shortened  by  pushing  the  sliding  part 

farther  in.  The  tube 
L  of  the  cornet  (Fig. 
246)  forms  several 
turns  or  convolu- 
tioji§j___w|^ich  may 
either  be  included 

or  cut  off  from   theremamder_of  the  tube  hy~ mea;ns~Qf~pi^tmisT 
d^c,  thus  varying  the  length  of  the  vibrating  column  of  air. 

The  bugle  (Fig.   247),  the  trumpet,  and  the  coaching  horn  are 
without  any  device  for  altering  the  length  of  the  air  column;  hence 


b   c 


FIG.  245.  —  The  Trombone. 


FIG.  247.  —  Bugle. 


FIG.  246.— 


their  notes  are  limited  to  the  fundamental  and  the  complete  series 
of  overtones.  The  different  notes  are  produced  by  varying  the  force 
of  the  breath  and  the  tension  of  the  lips. — -^ 

The  flute  and  the  clarinet  are  provided  with  holes  at  different 
distances,  which  are  closed  by  the  fingers  or  by  keys.  An  antinode 
is  produced  at  an  open  hole.  This  modifies  the  division  of  the  air 
column  into  vibrating  segments,  and  hence  determines  the  note. 


PROBLEMS 

1.  Compute  the  length  of  a  stopped  pipe  that  sounds  Ci  ( =  32)  at  20°  C., 
making  no  allowance  for  the  diameter. 

2.  Compute  the  length  of  an  open  pipe  that  sounds  c'  (=  256);    also 
the  length  of  one  that  sounds  cv  (=  4096). 

293.  The  Human  Voice.  —  The  special  organ  of  the  voice  is  the 
larynx,  the  front  half  of  which  is  shown  in  Fig.  248.  It  is  a  short 
tubular  box,  situated  at  the  upper  end  of  the  windpipe  and  opening 


SYMPATHETIC  AND   FORCED   VIBRATIONS       355 


248.  —  Front  Half 
of  the  Larynx,  Viewed 
from  the  Rear. 


above  into  the  back  part  of  the  mouth,  just  below  the  base  of  the 
tongue.  The  walls  of  the  larynx  consist  of  movable  plates  of  cartilage, 
joined  together  by  muscles.  The  large  angu- 
lar plate  Th  causes  the  prominence  in  the 
neck  known  as  "Adam's  apple."  The  leaf- 
shaped  plate  Ep  is  the  epiglottis,  which  folds 
back  as  a  lid  in  the  act  of  swallowing. 
Within  the  larynx  two  ridges  of  elastic  tissue 
project  toward  each  other  from  opposite  sides. 
Their  sharp  free  edges,  V,  extending  from 
front  to  rear  across  the  middle  of  the  larynx, 
are  the  vocal  cords.  Strictly  speaking,  they 
are  bands  rather  than  cords.  In  quiet  breath- 
ing the  vocal  cords  are  relaxed,  leaving  a  free 
passage  for  the  air  to  and  from  the  lungs,  as 
shown  in  the  top  view  of  Fig.  249.8.  In 
speaking,  the  cords  are  drawn  toward  each  FIG. 
other  by  muscular  action,  leaving  only  a 
narrow  slit  between  their  parallel  edges 
(Fig.  249^1);  and  the  current  of  air  from  the  lungs  sets  them  in 
vibration. 

The  vocal  cords  alone  would  produce  only  a  very  feeble  sound; 
but  their  vibration  is  reenforced  and  modified  in  quality  by  the 
sympathetic  vibration  of  the  air  in  the  mouth  and  nasal  cavities, 
which  act  as  resonance  chambers.  The  natural  pitch  of  the  voice 
depends  upon  the  size  of  the  larynx,  which  is  largest  in  men,  smaller 
in  women,  and  smallest  in  children.  The  tension  of  the  vocal  cords 
can  be  varied  at  will  by  the  muscles  of  the  larynx,  thus  permitting  a 
variation  of  pitch  within  a  certain  range,  known  as  the  compass  of 
the  voice.  The  quality  of  the  voice  depends  primarily  upon  the 
natural  size  and  form  of  the.  larynx  and  the  different  resonance 
cavities;  but  it  also  depends  to  a  great  extent  upon  the  muscular 
control  of  the  size,  shape,  and  opening  of  the  cavities,  and  may 
therefore  be  improved  by  cultivation. 

The  modulation  of  the  voice  into  speech  is  effected  by  means  of 
the  soft  palate,  tongue,  cheeks,  teeth,  and  lips.  By  various  move- 
ments of  these  parts  the  air  current  is  sometimes  interrupted,  as  in 
sounding  the  letters  p,  b,  t,  d,  k,  g,  and  sometimes  forced  through 
a  narrow  passage  in  the  mouth,  as  in  sounding  /,  v,  s,  z,  thus  adding 


356 


SOUND 


new  sounds  to  those  originated  by  the  vocal  cords.  Some  sounds,  as 
p,  t,  k,  /,  s,  originate  in  the  mouth,  the  vocal  cords  being  silent. 
In  whispering  none  of  the  sounds  are  voiced.  The  mouth  cavity 
responds  to  different  components  of  the  complex  vibrations  of  the 
vocal  cords,  according  to  its  size  and  shape  and  the  position  of  the 
lips.  This  gives  rise  to  the  different  vowel  sounds,  which  differ  from 


Epiglottis 
False  Vocal  Cords 

True  Vocal  Cords 


FlG.  249. 

one  another  only  in  quality.  Note  the  progressive  change  in  the 
shape  of  the  mouth  and  lips  in  repeating  the  vowels  e,  a,  ah,  aw, 
oh,  oo.  "  Helmholtz  analyzed  the  vowel  sounds  into  their  constituent 
tones,  imitated  these  tones  with  tuning  forks,  and  then,  by  recombin- 
ing  the  sounds  of  the  tuning  forks,  he  succeeded  in  reproducing  the 
vowel  sounds  artificially." 

294.  The  Ear.  —  The  ear  consists  of  three  parts:  the  external 
ear,  the  middle  ear  or  tympanum,  and  the  internal  ear  or  labyrinth. 
The  sensation  of  sound  originates  in  the  internal  ear;  the  other  parts 
merely  serve  the  mechanical  purpose  of  receiving  and  transmitting 
vibrations. 

The  external  ear  consists  of  the  expanded  part  on  the  exterior 
of  the  head  and  the  passage  leading  inward  from  it.  This  passage  is 
a  little  over  an  inch  in  length,  and  is  closed  at  its  inner  end  by  a  thin 
membranous  partition,  commonly  known  as  the  eardrum.  Its  true 
name  is  the  tympanic  or  drum  membrane.  This  membrane  is  kept 
under  tension  by  a  small  muscle,  which  pulls  it  in  at  the  center,  giving 
it  a  conical  shape.  The  expanded  part  of  the  external  ear  concen- 
trates the  sound  waves  and  directs  them  into  the  passage,  at  the  inner 
end  of  which  they  beat  upon  the  drum  membrane,  setting  it  into 
forced  vibration. 

The  middle  ear,  or  tympanum  (P,  Fig.  250,  and  Fig.  251),  is  an  ir- 
regular cavity  in  the  temporal  bone,  separated  from  the  external  ear 


SYMPATHETIC    AND    FORCED    VIBRATIONS      357 

by  the  drum  membrane.     It  is  connected  with  the  back  part  of  the 
mouth  (pharynx)  by  the  Eustachian  tube,  R  (Fig.  250),  which  allows 


FIG.  250.  —  Section  of  the  Right  Ear. 

air  to  pass  between  the  throat  and  the  middle  ear,  thus  equalizing  the 
pressure  on  the  two  sides  of  the  drum  membrane.  A  chain  of  three 
small  bones,  m,  i,  s,  extends  across  the  middle  ear  from  the  drum  mem- 
brane to  another  membrane  which  covers  a  small  opening  in  the  bony 
partition  between  the  middle  ear  and 
the  internal  ear.  By  means  of  this 
chain  the  vibrations  of  the  drum  mem- 
brane are  transmitted  to  the  liquid 
which  fills  the  internal  ear. 

The  internal  ear,  or  labyrinth, 
occupies  several  chambers  and  tubes 
hollowed  out  in  the  temporal  bone. 
The  middle  chamber,  called  the  vesti- 
bule, V  (Fig.  250),  is  just  back  of  the 
middle  ear.  Behind,  it  opens  into 
three  semicircular  canals,  one  of  which 
is  shown  at  b  in  the  figure;  in  front,  it  opens  into  a  spirally  coiled 
tube,  S,  called  the  cochlea,  from  its  resemblance  to  the  shell  of  a 


FIG.  251.  — The  Middle  Ear. 


358 


SOUND 


snail.  These  bony  chambers  and  tubes  contain  membranous  cham- 
bers and  tubes,  in  certain  parts  of  which  the  fibers  of  the  auditory 
nerve,  A,  end.  The  space  within  and  without  the  membranous 
parts  is  filled  with  a  watery  liquid.  The  function  of  the  semicircular 
canals  seems  to  be  distinct  from  that  of  hearing. 

The  cochlea  forms  a  spiral  of  two  and  a  half  turns,  and  contains 
a  bony  ledge,  5  (Fig.  250),  which  projects  into  it  from  the  axis.  It  is 
divided  into  two  channels,  A  and  B  (Fig.  252),  by  a  membranous 
tube,  C,  called  the  canal  of  the  cochlea, 
which  is  attached  at  its  inner  edge  to  the 
spiral  ledge  and  at  its  outer  surface  to  the 
wall  of  the  cochlea.  The  channels  and 
the  canal  are  filled  with  a  watery  liquid  which 
receives  and  transmits  vibrations  from  the 
middle  ear.  The  lower  side  of  the  canal, 
mb,  called  the  basilar  membrane,  consists  of 
thousands  of  delicate  fibers,  which  extend 
across  from  the  ledge  to  the  outer  wall. 
This  membrane  carries  upon  its  inner  sur- 
face a  remarkable  cellular  structure,  known  as  the  organ  of  Corti,  in 
which  the  filaments  of  the  auditory  nerve  terminate.  The  membrane 
gradually  widens  from  its  lower  to  its  upper  end,  as  shown  dia- 
grammatically  in  Fig.  253,  and  is  supposed  to  vibrate  sympathetically 
in  transverse  segments  (as  a,  b,  c)  in  re- 
sponse  to  the  vibrations  of  the  surrounding 
liquid,  the  rate  of  each  segment  depending 
in  part  upon  its  length.  Thus  when  a 


FIG.  252.  —  Cross-section 
of  the  Cochlea. 


FIG.  253. 


number  of  vibrations  of  different  frequencies,  such  as  constitute  an 
ordinary  musical  sound,  are  transmitted  to  the  cochlea,  they  (it  is 
supposed)  throw  into  sympathetic  vibration  those  parts  of  the  basilar 
membrane  which  have  the  same  natural  rate,  and  the  filaments 
of  the  auditory  nerve  ending  in  the  overlying  parts  of  the  organ  of 
Corti  are  stimulated.  According  to  this  theory,  the  recognition 
of  pitch  is  due  to  the  stimulation  of  different  nerve  fibers  by  differ- 
ent rates  of  vibration. 


CHAPTER  X 
LIGHT 

I.  NATURE  AND  TRANSMISSION  OF  LIGHT 

295.  Theories  of  Light.  —  Light  is  the  physical  cause  of 
the  sensations  of  brightness  and  color  which  we  receive 
through  the  eye.  These  sensations  alone  yield  no  informa- 
tion concerning  the  nature  of  their  cause.  What  light  really 
is  and  how  light  of  one  color  differs  in  itself  from  light  of 
another  color  are  theoretical  questions  upon  which  the 
scientific  world  was  long  in  coming  to  agreement. 

Two  wholly  different  views  were  current  during  the  lat- 
ter part  of  the  seventeenth  century.  Newton  adopted 
and  developed  the  emission  or  corpuscular  theory,  first 
proposed  by  the  philosophers  of  ancient  Greece.  Accord- 
ing to  this  theory  a  luminous  body  emits  streams  of  minute 
particles,  called  corpuscles,  which  travel  in  straight  lines, 
and  on  entering  the  eye  cause  vision  by  their  impact  on 
the  retina.  Opposed  to  this  was  the  undulatory  or  wave 
theory,  formulated  by  the  noted  Dutch  physicist,  Chris- 
tian Huygens,  in  1678.  According  to  this  view,  light  is  a 
periodic  disturbance  transmitted  as  a  wave  motion  through 
a  subtile  medium,  called  the  ether,  which  fills  all  space  not 
occupied  by  the  molecules  of  ordinary  matter.  (The  pupil 
should  carefully  review  the  outline  of  this  theory  presented 
in  Arts.  194-199,  under  the  subject  of  Radiation.) 

The  optical  phenomena  then  known  afforded  no  deci- 

359 


360  LIGHT 

sive  evidence  for  or  against  either  theory;  but  Newton's 
recognized  leadership  in  the  scientific  world  carried  the 
decision  in  favor  of  his  corpuscular  theory,  which  was 
very  generally  accepted  for  more  than  a  hundred  years. 
At  the  beginning  of  the  nineteenth  century  the  subject 
was  again  taken  up  by  able  physicists  in  France  and  Eng- 
land. By  a  series  of  remarkable  experiments  and  mathe- 
matical investigations,  new  facts  were  established  which 
only  the  wave  theory  could  explain,  and  this  theory  is  now 
accepted  by  all.  A  discussion  of  the  evidence  lies  beyond 
the  scope  of  elementary  physics;  but  we  are  at  liberty  to 
accept  the  conclusions  and  to  make  such  use  of  them  as 
our  purpose  requires. 

296.  Properties  of  the  Ether.  —  Since  the  ether  does 
not  affect  our  senses  directly,  its  properties  can  only  be 
inferred  from  optical  and  electrical  phenomena.  It  has 
inertia  or  mass,  since  time  is  required  for  a  disturbance 
to  travel  through  it.  It  is  apparently  a  continuous  sub- 
stance without  structure,  not,  like  ordinary  matter,  an 
assemblage  of  individual  particles  with  spaces  between. 

It  has  elasticity,  otherwise  it  would  be  incapable  of  trans- 
mitting a  disturbance.  Certain  optical  phenomena  show 
that  its  vibrations  are  transverse,  i.e.  at  right  angles  to 
the  direction  of  propagation  of  the  waves.  Its  elasticity 
must  therefore  be  similar  to  the  elasticity  of  form  of  solids, 
which  enables  them  to  recover  from  distortion,  rather  than 
the  elasticity  of  volume  of  liquids  and  gases,  which  enables 
them  to  recover  from  compression.  Yet,  although  the 
ether  has  a  certain  rigidity,  like  a  solid,  ordinary  matter 
passes  through  it  "with  as  much  ease  as  a  hand  with  fingers 
separated  can  move  through  air."  It  has  been  compared 
to  "an  all-pervading  impalpable  jelly." 


NATURE  AND  TRANSMISSION  OF  LIGHT        361 

297.  Light  and  Other  Radiant  Energy.  —  According  to 
the  wave  theory  the  vibrating  molecules  of  all  bodies  at 
all  temperatures  disturb  the  surrounding  ether,  each 
molecule  being  the  center  of  a  wave  motion  in  the  ether, 
just  as  a  sounding  body  is  the  center  of  a  wave  motion  in 
the  air;  and  as  the  energy  of  a  sounding  body  is  gradually 
imparted  to  the  sound  waves,  so  the  heat  energy  of  bodies 
is  imparted  to  the  ether  waves.  This  energy  of  the  ether 
is  called  radiant  energy. 

The  different  molecules  of  a  solid  vibrate  at  very  unequal 
rates,  even  when  the  temperature  is  uniform  throughout; 
and  the  faster  the  rate  the  shorter  are  the  ether  waves,  as 
in  -the  case  of  sound.  At  any  temperature,  therefore,  a 
body  radiates  waves  of  many  different  lengths;  but  as 
the  temperature  rises  the  average  rate  of  the  molecules 
increases  and  shorter  waves  are  produced. 

The  human  eye  is  sensitive  only  to  ether  waves  of  cer- 
tain lengths;  hence  the  distinction  between  light  or  visible 
radiation  and  invisible  or  non-luminous  radiation.  Invis- 
ible radiation  consists  of  ether  waves  which  are  too  long 
or  too  short  to  affect  the  eye.  In  the  older  language  of 
the  subject  these  longer  waves  were  called  "heat  waves/ ' 
"heat  rays,"  or  "radiant  heat,"  since  the  heating  effect  of 
radiant  energy  is  due  mainly  to  the  absorption  of  such 
waves.  We  shall  call  them  infra-red  waves  (waves  below 
the  red);  and  the  waves  which  are  too  short  to  affect  the 
eye  we  shall  call  ultra-violet  waves  (waves  beyond  the  vio- 
let). The  reasons  for  these  terms  will  appear  later. 

With  few  exceptions  bodies  emit  only  infra-red  waves 
unless  they  are  very  hot.  A  mass  of  iron  becomes  lumi- 
nous at  about  525°  C.,  at  which  temperature  it  emits  a  dull 
red  light.  As  its  temperature  rises,  the  light  grows  stronger 
and  changes  in  color  through  bright  red,  orange,  and  yel- 


362  LIGHT 

low,  at  last  becoming  white.  The  infra-red  radiation  is 
also  stronger  at  the  higher  temperature,  as  is  shown  by  its 
greater  heating  power. 

298.  The  Velocity  of  Light.  —  The  velocity  of  light  in 
empty  space,  in  air,  and  in  other  transparent  media  has 
been  determined  with  great  accuracy  from  astronomical 
observations  and  by  different  experimental  methods.     In 
round  numbers,  the  velocity  in  a  vacuum  is  186,000  mi. 
per  second,  —  a  velocity  sufficient  to  encircle  the  earth 
seven  and  one-half  times  in  one  second.     In  air  it  is  only 
very  slightly  less,  in  water  it  is  about  three  fourths  as  great, 
and  in  carbon  bisulphide  about  three  fifths  as  great.    In  all 
transparent  solids  and  liquids  it  is  considerably  less  than 
in  air  and  other  gases. 

Light  reaches  us  from  the  sun  in  8  m.  and  20  sec.  An  express 
train,  traveling  at  the  rate  of  42  mi.  per  hour,  would  require  254  yr. 
for  the  journey.  Yet  inconceivably  great  as  this  distance  is,  it  is 
insignificant  in  comparison  with  that  of  the  fixed  stars,  the  nearest 
of  which  is  so  far  away  that  light  from  it  requires  3.5  yr.  to  reach 
us.  Light  from  the  north  star  requires  about  50  yr.  for  the  journey. 
The  vast  majority  of  the  stars  are  at  still  greater  distances. 

The  early  attempts  of  Galileo  and  others  to  determine  the  velocity 
of  light,  by  methods  similar  to  those  employed  for  sound,  only  proved 
that,  if  time  was  actually  required  for  light  to  travel,  it  was  too 'short 
to  be  detected  by  such  means.  The  first  determination  of  the 
velocity  of  light  was  made  by  Roemer,  a  Danish  astronomer,  in  1675. 
His  computation  was  based  on  data  relating  to  the  eclipses  of  one  of 
Jupiter's  moons.  Between  1849  and  1882  the  velocity  was  determined 
experimentally  by  several  investigators  in  France  and  the  United 
States.  An  account  of  these  notable  observations  and  experiments 
is  to  be  found  in  larger  works. 

299.  The  Transmission  of  Light.  —  Light  travels  through 
many  substances,  such  as  air,  glass,  water,  etc.;  but  in  all 
such  cases  it  consists  of  waves  in  the  ether  which  fills  the 


NATURE  AND   TRANSMISSION  OF  LIGHT        363 

intermolecular  spaces  within  the  substance.  Ordinary 
matter  is  at  best  more  or  less  of  a  hindrance  to  the 
transmission  of  light,  and  affects  its  behavior  in  various 
ways,  especially  at  places  where  the  light  passes  from  one 
substance  into  another.  But  within  any  uniform,  transpar- 
ent medium  it  travels  by  straight  paths,  as  in  empty  space. 
The  straight  course  of  a  sunbeam,  as  seen  in  a  darkened 
room,  is  a  good  example.  The  transmission  of  light  in 
straight  lines  is  also  shown  in  the  formation  of  shadows. 

A  body  of  light  radiating  from  a  point  and  limited  in 
direction  takes  the  form 
of  a  cone,  and  is  called  a 
diverging  cone  or  pencil 
of  light.  For  example, 
when  light  from  a  lumi- 
nous point,  I  (Fig.  254), 
falls  upon  a  screen,  AB, 
in  which  there  is  an 
opening,  O,  a  cone  of 
light,  CID,  passes 
through  the  opening. 
(The  figure  represents  a  plane  section  through  the  screen, 
the  hole,  and  the  light,  the  section  of  the  cone  of  light 
A  being  a  triangle  and  the 

section  of  a  light  wave  a 
E  c  circle.) 

ilU  ]  1 1 1 1 1 1 1       When  light  radiates  from 

Lllllllll'll  •  i     j  •        i 

D  a  point  at  a  relatively 
great  distance,  any  small 
area  of  a  wave  front  is 

FIG.  255.  — Section  of  a  Beam  of  Light,      sensibly    plane    (Fig.    255), 

and  a  limited  body  of  the  light,  CEFD,  such  as  passes 
through  a  small  opening,  has  the  form  of  a  cylinder  and 


FIG.  254.  — Section  of  a  Cone  of  Light. 


364  LIGHT 

is  called  a  beam  of  light.  A  very  slender  cone  or  beam 
of  light  is  commonly  called  a  ray  of  light.  In  another 
sense  a  ray  is  not  light  at  all,  but  merely  a  line  along 
which  some  point  of  a  wave  front  travels.  Rays  in 
either  sense  are  perpendicular  to  the  wave  fronts. 

300.    Shadows  and  Eclipses.  —  In  common  speech  the 
space  from  which  direct  light  from  any  source  is  wholly 

or  partly  cut  off  by  an  opaque 
body  is  called  a  shade,  and  the 
dark  area  upon  any  surface  which 
intercepts  the  shade  is  called  a 
shadow.  In  this  sense  a  shade 
occupies  the  three  dimensions  of 
space  and  a  shadow  only  two. 
As  the  terms  are  used  in  physics, 
the  darkened  space  of  three  di- 
mensions is  the  shadow,  and  the 
darkened  area  upon  any  surface 

FIG.    256.  —  Longitudinal    Sec-  *: 

don  of  a  shadow  Consisting   which   intercepts  the   shadow  is 

of  Umbra  only.  called    &    sectjon     Qf    tne    shadow. 


The  formation  of  shadows  can  be  understood  only  when  the 
thought  is  centered  on  the  shadow,  in  the  latter  sense  of  the 
term.  Sections  of  the  shadow,  taken  in  any  direction,  serve 
merely  as  an  aid  to  the  understanding  of  the  shadow  itself. 

When  the  source  of  the  light  is  so  small  that  it  may  be 
regarded  as  a  single  luminous  point,  P  (Fig.  256),  a  cone  of 
this  light  is  intercepted  by  any  opaque  body,  and  the  shadow 
of  the  body  occupies  the  portion  of  this  conical  space, 
which  lies  on  the  side  opposite  to  the  source.  This  shadow 
space  extends  indefinitely,  with  constantly  increasing  cross- 
section. 

In  any  actual  case  the  source  of  light  is  more  than  a 


NATURE   AND   TRANSMISSION  OF  LIGHT        365 

point,  and  its  greater  or  less  size  affects  the  character  of 
the  shadow.  In  the  case  shown  in  Fig.  257  the  opaque 
body  ab  is  slightly  larger  than  the 
candle  flame  mn.  The  light  is  entirely 
cut  off  from  the  space  cabd,  which 
really  extends  to  an  indefinite  distance. 
This  total  shadow  or  umbra  is  sur- 
rounded by  a  space  which  receives 
light  from  some  part  of  the  source, 
but  not  from  the  whole  of  it.  This 
is  the  partial  shadow  or  penumbra, 
and  is  represented  in  the  section  by 
eac  and  fbd.  The  penumbra  merges 
imperceptibly  into  fully  illuminated  space  at  its  outer 
surface;  its  inner  boundary  is  more  sharply  denned. 

When  the  source  of  light  is  larger  than  the  body  which 
casts  the  shadow,  the  umbra  ends  in  a  point  or  a  line  at  a 
definite  distance,  beyond  which  there  is  only  penumbra. 
The  shadows  of  the  planets  and  their  satellites  are  inter- 
esting examples.  The  umbra  is  a  long  cone,  extending  out 
into  space  away  from  the  sun  and  terminating  in  a  point. 

Solar  Eclipses.  —  In  Fig.  258,  S  represents  the  sun,  E  the  earth, 
Mi  the  moon  at  new  moon,  and  MZ  at  full  moon.  On  account  of  the 


m 


FIG.  257.— Longitudinal 
Section  of  a  Shadow. 
cabd,  Umbra;  eac  and 
bdf,  Penumbra. 


FIG.  258.  — Eclipses  of  the  Sun  and  Moon. 

varying  distances  of  the  sun  and  the  moon  from  the  earth,  the  moon's 
shadow  (the  umbra)  sometimes  reaches  the  earth  and  is  sometimes 
too  short  to  do  so.  Its  cross-section  is  never  more  than  167  mi. 
wide  at  the  earth's  surface.  Within  the  umbra  the  sun  is  totally 
eclipsed;  within  the  penumbra,  which  covers  a  much  larger  area, 


366  LIGHT 

the  eclipse  is  partial.  An  eclipse  of  the  sun  can  occur  only  at  new 
moon,  and  then  only  when  the  moon  passes  exactly  in  line  between 
some  portion  of  the  sun  and  the  earth.  This  occurs  from  two  to  five 
times  in  a  year. 

Lunar  Eclipses.  —  When  the  moon  passes  wholly  within  the  earth's 
shadow,  it  is  totally  eclipsed;  when  only  one  side  of  it  passes  through, 
the  eclipse  is  partial.  There  is  no  perceptible  dimming  of  the  moon 
within  the  penumbra  until  it  is  very  near  the  umbra.  An  eclipse 
of  the  moon,  either  total  or  partial,  is  of  course  visible  to  half  the  earth 
at  the  same  time.  Since  the  moon  shines  only  by  reflected  sunlight, 
it  would  be  invisible  when  totally  eclipsed  if  it  were  not  for  the  fact 
that  some  light  is  bent  out  of  its  course  (refracted)  into  the  shadow 
in  passing  through  the  earth's  atmosphere.  The  moon  is  thus  illu- 
minated with  a  dull,  copper-colored  light.  A  lunar  eclipse  can  occur 
only  at  full  moon;  but  the  moon  generally  escapes  the  shadow  by 
passing  to  one  side  or  the  other  of  it.  The  number  of  lunar  eclipses 
in  a  year  varies  from  none  to  three. 

301.  Why  Light  Travels  in  Straight  Lines.  —  The  fact  that  light 
does  not  appear  to  travel  round  obstacles  or  to  spread  after  passing 
through  openings,  as  water  waves  and  sound  waves  do,  was  regarded 
by  Newton  and  his  followers  as  a  fatal  objection  to  the  wave 
theory.  On  the  other  hand,  this  behavior  is  just  what  we  should 
expect  if  light  consisted  of  flying  particles.  More  than  a  century 
elapsed  before  it  was  discovered  that  this  apparent  difference  in  the 
behavior  of  light  and  sound  is  due  to  the  enormous  difference  in  their 
wave  lengths.  Sounds  of  ordinary  pitch  vary  in  wave  length  from 
one  to  ten  feet;  light  waves  vary  from  33,000  to  64,000  to  the 
inch.  Sound  and  light  behave  in  a  similar  manner  when  they  pass 
through  openings  or  encounter  obstacles  of  the  same  relative  size 
in  comparison  with  their  wave  lengths.  Pins  and  pinholes  bear 
about  the  same  relation  to  light  waves  that  mountains  and  valleys 
do  to  sound  waves.  Now  it  is  found  that  an  intervening 
range  of  hills  affords  complete  protection  from  the  most  violent 
disturbances  of  the  air,  as  in  the  explosion  of  powder-magazines; 
i.e.  the  hills  cast  a  sound  shadow.  Sounds  of  high  pitch,  due  to 
very  short  waves,  fail  to  pass  round  smaller  obstacles,  such  as  large 
buildings. 

On  the  other  hand,  when  light  passes  through  an  opening  that  is 


NATURE  AND   TRANSMISSION  OF   LIGHT        367 

not  large  in  comparison  with  the  wave  length,  it  spreads  out  into  the 
region  that  is  ordinarily  occupied  by  the  shadow.  A  simple  example 
of  these  effects  is  presented  by  looking  through  a  handkerchief,  held 
close  to  the  face,  at  a  brightly  illuminated  pinhole  or  a  narrow  slit 
about  the  width  of  a  pin,  or  at  the  filament  of  an  incandescent  electric 
lamp.  (Try  it.)  The  pinhole  or  slit  may  be  made  in  a  piece  of 
cardboard,  and  illuminated  by  holding  it  in  front  of  a  lamp  or  a  gas 
jet.  The  light  spreads  out  in  different  directions  in  passing  through 
the  narrow  spaces  between  the  threads  of  cloth,  causing  the  slit  to 
look  like  a  number  of  parallel  slits,  and  the  hole  like  a  square  pattern 
of  many  holes. 

A  further  illustration  of  the  effect  of  wave  length  when  obstacles 
are  encountered  is  afforded  by  water  waves.  Large  water  waves 
pass  round  a  pile  or  other  comparatively  small  obstacle;  but  ripples 
are  effectually  stopped,  passing  the  object  on  each  side  without 
reuniting,  and  leaving  a  well-defined  region  of  no  disturbance  or 
ripple  shadow. 

302.  Pinhole  Images.  —  When  light  from  any  luminous 
or  brightly  illuminated  object  falls  upon  a  screen  after 
passing  through 
a  minute  open- 
ing, such  as  a 
pinhole,  it  forms 
upon  the  screen 
an  inverted  im- 
age of  the  object 
(Fig.  259). 
Every  point  on  the  surface  of  the  object  radiates  or 
reflects  light  in  all  directions.  A  slender  cone  of  this  light 
from  each  point  of  the  source  passes  through  the  pinhole 
^and  illuminates  a  small  spot  on  the  screen,  of  the  same 
shape  as  the  opening.  Since  these  spots  have  the  same 
relative  positions  as  the  corresponding  points  of  the  object, 
and  are  illuminated  by  light  of  the  same  color  as  those 
points,  they  unite  into  an  image  which  reproduces  the  form 


FIG.  259.  —  Formation  of  a  Pinhole  Image. 


368  LIGHT 

and  color  of  the  object.  The  inversion  of  the  image  is 
due  to  the  crossing  of  the  cones  of  light  at  the  pinhole. 
If  the  pinhole  is  very  small,  the  image  is  quite  sharply 
denned,  but  faint.  With  a  larger  opening,  the  image  is 
brighter  but  poorly  defined  or  blurred,  for  the  light  from 
each  point  of  the  object  now  covers  a  larger  spot  on  the 
screen,  and  these  spots  overlap  more  and  more  as  their 
size  increases. 

II.    INTENSITY  OF  ILLUMINATION.    CANDLE  POWER 

303.  Relation  Between  Intensity  of  Illumination  and 
Distance.  —  Experience  teaches  that  the  intensity  of  the 
light  from  any  source  decreases  with  increasing  distance 
from  the  source.     We  hold  a  book  nearer  a  lamp  to  get  a 
stronger  illumination  of  the  page.     The  relation  between 

intensity  and  distance  is  the  same 
for  light  as  for  sound  (Art.  266); 
i.e.  the  intensity  varies  inversely  as 
the  square  of  the  distance. 

For  example,  if  a  cone  of  light  from 
FIG.  260.  —  Relation  of  Cross- 

.'-»..  any  point  covers  i  sq.  cm.  at  a  certain 

section  and  Distance. 

distance,   A    (Fig.   260),  it  will  spread 

over  4  sq.  cm.  at  twice  that  distance,  and  over  9  sq.  cm.  at  three 
times  that  distance;  hence  the  intensity,  or  quantity  of  light  per 
unit  area,  is  \  as  great  at  twice  the  distance,  \  as  great  at  three 
times  the  distance,  etc. 

304.  Illuminating  Power   of   Sources   of  Light.  —  The 
rate  at  which  a  luminous  body  gives  out  light  is  called  its 
illuminating  power.     At  a  given  distance  from  any  source 
of  light  the  intensity  of  illumination  produced  by  it  is 
proportional  to  its  illuminating  power. 

The  unit  of  illuminating  power  varies  slightly  in  different  countries. 
In  England  and  America  it  was,  until  recently,  the  light  emitted  by 


INTENSITY  OF  ILLUMINATION  369 

a  sperm  candle  weighing  one  sixth  of  a  pound  and  burning  1 20  grains 
per  hour.  This  unit  is  called  the  candle  power.  An  incandescent 
electric  lamp  of  sixteen  candle  power  gives  out  sixteen  times  as  much 
light  as  such  a  candle.  The  present  unit  in  England,  France,  and  the 
United  States  (adopted  in  1909)  is  known  as  the  international  candle 
power.  It  is  1.6  %  less  than  the  old  unit.  Since  the  light  of  a 
candle  fluctuates  considerably,  other  sources  have  been  adopted  as 
standards.  In  America  the  power  of  gas  lights  is  determined  by  com- 
parison with  a  standard  pentane  gas  flame,  and  electric  lights  are 
compared  with  standardized  incandescent  lamps. 

305.  Photometry  deals  with  the  comparison  and  measure- 
ment of  the  illuminating  powers  of  different  sources  of 
light.  Any  apparatus  by  means  of  which  such  measure- 
ments are  made  is  called  a  photometer.  We  can  form  no 
reliable  estimate  of  the  relative  brightness  of  unequally 
illuminated  surfaces ;  but  we  are  able  to  judge  with  consid- 
erable accuracy  whether  two  adjacent  parts  of  the  same 
surface  are  equally  illuminated,  and  all  methods  of  compari- 
son are  based  on  this  principle. 

Now  when  two  sources  of  light  are  placed  at  such  dis- 
tances that  they  illuminate  the  same  surface  equally,  the 
ratio  of  their  illuminating  powers  is  given  by  the  square 
of  the  ratio  of  their  respective  distances  from  the  surface. 
This  follows  from  the  relation  between  intensity  and  dis- 
tance. For  example,  if  one  source  is  four  times  as  strong 
as  the  other,  the  intensity  of  its  light  is  four  times  that  of 
the  other  at  the  same  distance,  and  is  equal  to  that  of  the 
other  at  twice  as  great  a  distance. 

There  are  several  forms  of  photometers;  but  either  of 
the  following  will  serve  to  illustrate  the  general  principle. 
The  shadow  photometer  (Fig.  261),  devised  by  Count 
Rumford,  consists  essentially  of  a  rod  and  a  screen,  sup- 
ported in  a  vertical  position,  and  a  scale  for  measuring 
distances.  The  rod  casts  two  shadows  on  the  screen, 


37° 


LIGHT 


corresponding  to  the  two  sources  of  light;  and  each  source 
illuminates  the  shadow  due  to  the  other.  The  distances 
are  so  adjusted  that  the  shadows  appear  equally  dark. 
They  are  then  equally  illuminated,  each  by  one  source 


\ 


FIG.  261.  —  Rumford's  Photometer. 

only;  and  the  ratio  of  the  illuminating  powers  of  the  two 
sources  is  then  given  by  the  square  of  the  ratio  of  their 
respective  distances  from  the  screen.  The  comparison  of 
shadows  requires  a  darkened  room,  the  darker  the  better, 
although  a  considerable  amount  of  diffused  light  does 
not  greatly  affect  the  result. 

The  essential  part  of  a  Bunsen  photometer  is  a  vertical  screen  of 
white  paper,  having  a  translucent  spot  made  by  applying  a  drop  of 
hot  paraffin  (Fig.  262).  Since  the  spot  transmits  more  light  than 
the  rest  of  the  paper  and  reflects  less,  it  appears  dark  when  viewed 


FIG.  262.  —  The  Bunsen  Photometer. 

from  the  more  strongly  illuminated  side,  and  light  when  viewed  from 
the  less  strongly  illuminated  side.  When  both  sides  are  equally 
illuminated,  the  spot  nearly  disappears  and  presents  exactly  the 
same  appearance  on  both  sides.  The  adjustment  of  the  photometer 
consists  in  moving  the  screen  from  side  to  side  along  the  line  between 


INTENSITY  OF  ILLUMINATION  371 

the  two  lights,  until  the  position  is  found  in  which  the  two  sides  of 
the  spot  look  exactly  alike.  This  adjustment  is  more  accurately 
made  with  the  aid  of  two  small  mirrors,  placed  so  that  the  two  sides 
of  the  screen  can  be  seen  in  them  at  the  same  time.  For  accurate 
work  the  photometer  is  inclosed  in  a  box  painted  a  dull  black  on  the 
inside  or  surrounded  by  black  screens  or  curtains,  so  as  to  cut  off  all 
diffused  light. 

306.  Industrial  Photometry.  —  Thirty  or  forty  years  ago  the 
principal  artificial  sources  of  light  were  the  candle,  the  oil  lamp,  and 
the  ordinary  gas  jet.  The  electric  arc  lamp  had  just  begun  to  prove  its 
value  for  outdoor  lighting  in  1880.  Several  inventors,  led  by  Edison 
in  America  and  Swan  in  England,  were  at  that  time  developing  the 
incandescent  electric  lamp;  but  it  was  not  brought  to  the  point  of 
commercial  success  until  several  years  later.  Since  then  many  forms 
of  electric  lamps  (Arts.  455  and  456)  have  come  into  general  use,  and 
the  Welsbach  burner  has  largely  superseded  the  common  burner 
in  gas  lighting.  The  principal  advantages  gained  by  these  newer 
sources  of  light  are  better  quality  of  light,  greater  illuminating  power, 
and  greater  luminous  efficiency  (ratio  of  candle  power  to  the 
amount  of  gas  consumed  in  a  given  time).  The  Welsbach  light,  for 
example,  gives  from  four  to  five  times  as  much  light  per  cubic  foot 
of  gas  consumed  as  an  ordinary  gas  jet.  There  has  also  been  a 
marked  improvement  in  shades  and  reflectors  designed  to  throw 
the  light  in  useful  directions  (Art.  326). 

In  designing  and  manufacturing  lamps,  burners,  shades,  reflectors, 
etc.,  photometric  measurements  are  always  necessary  to  determine 
the  intensity  of  the  light  and  its  distribution  in  different  directions. 
A  room  is  set  apart  for  this  purpose,  and  equipped  with  photometers 
and  with  apparatus  for  measuring  the  amount  of  gas  or  the  electric 
power  consumed.  The  walls  and  ceiling  of  the  photometer  room  are 
painted  a  dull  black  to  avoid  errors  from  diffused  light,  and  all  possi- 
\>\e  precautions  are  taken  to  secure  accuracy.  Manufacturers  issue 
catalogues  giving  the  results  of  these  experimental  tests,  so  that  buyers 
may  be  able  to  select  the  lighting  equipment  that  best  suits  their 
needs. 

PROBLEMS 

1.  What  does  the  act  of  aiming  a  gun  assume  concerning  light? 
Explain. 


372 


LIGHT 


2.  The  shadow  of  a  flagpole  is  80  ft.  long  upon  the  ground  when  the 
shadow  of  a  vertical  stick  4  ft.  high  is  2.5  ft.     What  is  the  height  of  the 
flagpole? 

3.  Why  are  shadows  as  we  see  them  in  nature  not  perfectly  dark? 

4.  What  is  the  apparent   shape  of  the  moon  two  or  three  days  after 
new  moon?   at  the  first  quarter?   between  the  first  quarter  and  full  moon? 
Account  for  these  different  appearances. 

6.  Assuming  the  diameter  of  the  sun  to  be  866,000  mi.,  the  diameter 
of  the  earth  8000  mi.,  and  the  distance  between  the  sun  and  the  earth 
93,000,000  mi.,  find  the  length  of  the  umbra  of  the  earth's  shadow. 

6.  State  and  account  for  the  change  in  the  size,  brilliance,  and  sharp- 
ness of  outline  of  a  pinhole  image  (a)  when  the  screen  upon  which  it  is 
caught  is  moved  farther  from  the  opening;   (b)  when  the  size  of  the  opening 
is  increased.  i 

7.  What  determines  the  ratio  of  the  length  of  a  pinhole  image  to  the 
length  of  the  object? 

8.  In  what  ratio  does  the  illumination  upon  the  page  of  a  book  change 
as  the  book  is  moved  from  a  distance  of  2  m.  to  a  distance  of  60  cm. 
from  a  lamp? 

9.  An  incandescent  lamp  at  a  distance  of  150  cm.  and  a  standard  candle 
at  a  distance  of  21  cm.  equally  illuminate  the  screen  of  a  photometer.     Find 
the  candle  power  of  the  lamp. 

III.    REFLECTION  OF  LIGHT 

307.  Regular  and  Irregular  Reflection.  —  The  behavior 
of  light  after  reflection  from  mirrors  and  other  surfaces  can 


FIG.  2630.  —  Regular  Reflection. 


FIG.  2636.  —  Irregular  Reflection. 


be  studied  to  the  best  advantage  by  means  of  a  beam  of 
sunlight  admitted  into  a  darkened  room.  Chalk  dust  in 
the  air  makes  the  path  of  the  light  plainly  visible.  Under 


REFLECTION   OF  LIGHT  373 

such  conditions  it  will  be  observed  that  the  reflection  of  a 
beam  by  a  plane  mirror  merely  changes  its  direction  (Fig. 
2630).  It  is  still  a  beam  of  light.  This  is  a  case  of  regular 
reflection.  When  the  beam  falls  upon  a  sheet  of  paper, 
the  light  is  scattered  or  diffused  in  all  directions  by  the 
minute  irregularities  of  the  surface  (Fig.  2636).  This  is 
called  irregular  or  diffuse  reflection.  It  will  be  observed 
that  the  paper  is  brilliantly  illuminated  by  the  sunbeam 
and  can  be  distinctly  seen  from  all  parts  of  the  room;  but 
the  surface  of  a  mirror  is  nearly  invisible,  even  when  the 
eye  is  in  the  path  of  the  reflected  beam. 

Light  is  regularly  reflected  only  from  surfaces  (either 
plane  or  curved)  which  appear  smooth  even  under  the  micro- 
scope, such  as  the  surfaces  of  still  water,  glass,  polished 
metals,  and  ordinary  mirrors.  An  unpolished  surface  is 
made  up  of  minute  areas  inclined  at  all  angles  to  one  an- 
other, and  each  one  acts  as  a  separate  reflector.  Any  small 
spot  or  " point"  of  such  a  surface  includes  a  sufficient 
number  of  these  irregularities  to  reflect  light  in  all  direc- 
tions, and  thus  in  effect  becomes  a  new  source  of  light 
waves. 

The  light  received  from  the  sky  during  the  day  is  sunlight  that  has 
been  diffused  by  minute  particles  of  dust  in  the  air.  This  diffusion 
takes  place  principally  in  the  lower  regions  of  the  atmopshere,  where 
the  dust  particles  are  largest  and  most  numerous;  hence  the  sky  is 
much  darker  upon  high  mountains  than  at  low  altitudes.  Shadows 
cast  by  objects  in  the  sunlight  are  only  relatively  dark,  as  they  are 
generally  quite  strongly  illuminated  by  diffused  light  from  the  sky 
and  from  surrounding  objects. 

308.  Visibility  of  Objects.  —  Objects  which  are  not 
themselves  luminous  are  seen  only  by  the  light  which  they 
diffuse.  Regular  reflection  produces  images.  The  light 
appears  to  come  from  the  image  of  the  source,  not  from  the 


374  LIGHT 

reflecting  surface.     A  good  mirror  with  a  clean  surface  is 

nearly  or  quite  invisible,  —  a  fact  which  is  often  turned  to 

account  in  producing  stage  illusions. 

When  an  object  is  viewed  directly  (i.e.  without  the  aid 

of   mirrors    of    lenses),    the   light   by  which    each   point 

is  seen  comes  straight  from 
the  point  to  the  eye  as  a 
slender  diverging  cone,  and 
the  point  is  seen  in  its  true 
position  at  the  vertex  of 

FIG.  264.  —  The  Eye  Receives  a  Slender    ,  v  •  /-r^'          n    \ 

Cone  of  Light  from  each  Visible  Point.      thlS  COne   VFlS'    264)- 

309.  Laws  of  Reflection.  —  When  light  falls  upon  any 
surface  the  phenomenon  is  called  the  incidence  of  light, 
and  the  light  before  it  reaches  the  surface  is  called  inci- 
dent light.  In  Fig.  265  MN  represents  a  section  through 
a  mirror  whose  surface  is  perpendicular  to  the  plane  of  the 
paper;  CP'  represents  a  slender 
beam  (ray) ,  falling  upon  the  mirror 

** 

at   P    (called   the  point  of   inci- 
dence),   and    P'D    the    reflected 
beam;  and  PP'  is  the  perpendi-  FIG.  *6S. -Reflection,  i,  Angle 
cular  or  normal  to    the   reflecting        of  incidence;  r,  Angle  of 

..  Reflection. 

surface  at  the  point  of  incidence. 

The  angle  i  between  the  incident  beam  (or  ray)  and  the 
perpendicular  to  the  reflecting  surface  at  the  point  of  inci- 
dence is  called  the  angle  of  incidence,  and  the  angle  r 
between  the  reflected  beam  and  this  perpendicular  is  the 
angle  of  reflection. 

Direct  measurement  establishes  the  following  laws  of 
reflection:  (i)  The  angles  of  incidence  and  reflection  are 
equal;  and  (2)  they  are  in  the  same  plane.  Since  this  plane 
contains  the  normal,  it  is  always  perpendicular  to  the  re- 
flecting surface. 


REFLECTION  OF  LIGHT 


375 


These  laws  are  readily  accounted  for  by  considering  what  happens 
to  light  waves  during  reflection  from  a  plane  surface.  Let  6*  (Fig. 
266)  be  a  point  source  radiating  light  waves  which  fall  upon  the 
mirror  MN.  The  shortest  distance  from  ,5*  to  the  mirror  is  along  the 


FIG.  266.  —  Position  of  a  Point  Image. 

perpendicular  from  5  to  the  mirror,  or  SP',  hence  each  wave  strikes 
the  mirror  first  at  P,  and  its  reflection  begins  at  that  point.  While 
the  part  A  of  the  wave  APB  is  advancing  to  A'  and  the  part  B  to 
B',  the  part  at  P  would  advance  to  P',  if  the  mirror  were  not  there, 
and  the  wave  would  occupy  the  position  A' P' B' ';  but  with  the 
mirror  in  place,  the  part  at  P  is  reflected  and  travels  back  an  equal 
distance  PP"  in  the  same  time,  and  the  entire  wave  is  reflected  so  as 
to  occupy  the  position  A'P"Bf.  Now  from  the  geometry  of  the 
figure  it  is  evident  that  the  arcs  A' P' B'  and  A' P"B'  are  equal  arcs 
of  equal  circles;  hence  their  radii  are  equal.  But  the  radius  of 
A1 P'B'  is  P'S;  hence,  laying  off  an  equal  distance  P"I  along  the 
perpendicular,  we  find  7,  which  is  the  center  of  the  arc  A'P"B'. 
I  is  therefore  the  center  of  the  reflected  waves. 

This  means  that  the  reflected  light  behaves  in  all  respects  as  if  it 
radiated  from  a  source  at  7.     The  direction  of  all  reflected  rays  is 


376  LIGHT 

from  /  as  a  center;  e.g.  light  traveling  along  SB'  is  reflected  along 
B'C.  Further,  it  is  easily  proved  from  the  geometry  of  the  figure 
that  the  angle  of  incidence  i  is  equal  to  the  angle  of  reflection  r,  and 
these  angles  evidently  lie  in  the  same  plane  (the  plane  of  the  paper). 
(The  proof  that  angle  i  =  angle  r  is  left  to  the  pupil.  Remember 
that,  by  construction,  SI  and  QB'  are  each  perpendicular  to  M  N, 
that  PP"  =  PP',  and  that  SP'  =  IP".) 

310.  The  Image  of  a  Point  in  a  Plane  Mirror.  —  It  is 

found  by  experiment  that  the  image  of  a  point  in  a  plane 
mirror  (from  whatever  position  it  may  be  viewed)  is  on 
the  perpendicular  from  the  point  to  the  mirror,  and  is  as 
far  behind  the  reflecting  surface  as  the  point  is  in  front  of 
it.  Now  in  Fig.  266  SP'  =  IP"  and  PP'  =  PP";  hence 
SP'  —  PP'  =  IP"  —  PP",  or  SP  =  IP.  That  is,  the  cen- 
ter /  of  the  reflected  waves  is  on  the  perpendicular  from 
the  point  source  S  to  the  mirror,  and  is  as  far  behind  the 
mirror  as  the  source  is  in  front  of  it.  It  follows  that  the 
point  image  is  at  the  center  of  the  reflected  waves.  The  im- 
age is,  in  fact,  nothing  else  than  the  apparent  source  of  the 
reflected  light. 

In  looking  at  the  image  the  eye  receives  a  cone  of  light 
whose  vertex  is  at  the  image;  and  the  impression  received 
is  just  the  same  as  if  this  vertex  were  the  real  source.  The 
eye  takes  no  account  of  the  original  direction  of  the  light. 

In  all  diagrams  representing  optical  phenomena  the  actual  path 
of  light  is  represented  by  full  lines  and  the  apparent  path  by  dotted 
lines,  as  in  the  above  figure. 

311.  Image  of  a  Body  Object  Formed  by  a  Plane  Mir- 
ror. —  In  the  discussion  of  images  we  shall  call  an  object 
having  visible  form  and  dimensions  a  body  object,  to  dis- 
tinguish it  from  a  point  object  or  point  source  of  light. 
The  image  of  a  body  object  is  made  up  of  point  images  of  the 
corresponding  points  of  the  body;  i.e.  the  light  radiated  or 


REFLECTION  OF  LIGHT 


377 


diffused  from  each  point  of  the  body  falls  upon  the  mirror, 
is  reflected,  and  after  reflection  travels  from  the  direction 
of  its  point  image,  just  as  if  no  other  M 
points  and  no  other  light  were  there.  c, 

It  follows,  therefore,  that  the  line  join- 
ing any  point  of  a  body  and  the  image  of  j    ^ 


that  point,  formed  by  a  plane  mirror,  is  %-- — 

N 

FIG.  267. 


perpendicular  to  the  mirror  and  is  bisected 

by  it  (Fig.  267).  Images  formed  by  plane 

mirrors  are  evidently  of  the  same  size   as   the   objects. 

They  are  erect   (unless  the  mirror  is  horizontal,  as  the 

surface  of  still  water);  but  object  and  image  differ  as  the 

right  hand  differs  from  the  left. 

To  locate  such  an  image  in  a  diagram  perpendiculars  are  drawn 
from  a  sufficient  number  of  points  of  the  object  to  the  mirror,  and  ex- 
tended equal  distances  behind  it,  as  shown  in  the  figure.  The  ex- 
tremities of  these  perpendiculars  are  the  corresponding  points  of  the 
image.  To  construct  the  path  of  light  to  the  eye  for  any  point  of 
the  image,  as  ZX,  we  draw  a  line  from  that  point  to  the  eye,  and  from 
the  point  of  intersection  of  this  line  with  the  mirror  we  draw  a  line 
to  the  corresponding  point  of  the  object.  This  construction  makes 

the  angles  of  incidence  and  reflection 
equal  without  the  trouble  of  meas- 
uring them.  Such  diagrams  are  com- 
monly simplified  by  drawing  only  a 
single  line  to  represent  the  cone  of 
light  that  enters  the  eye  from  any 
point. 

312.  Images  by  Multiple  Reflec- 
tion. —  When  two  mirrors  are  placed 
facing  each  other,  either  parallel  or  at 
an  angle,  a  part  of  the  light  from  any  source  between  them  is  reflected 
from  one  to  the  other,  giving  rise  to  a  series  of  images.  Thus  with 
mirrors  AB  and  CD  (Fig.  268),  placed  as  shown,  three  images  of 
the  point  source  O  are  formed  as  follows:  A  single  reflection  from 


FIG.  268. 


378 

AB  forms  the  image 


LIGHT 


FIG.  269.  —  Kaleidoscopic   Pattern   Formed 
by  Three  Mirrors  at  Angles  of  60°. 


Some  of  this  reflected  light  falls  upon  the 
mirror  CD,  and  is  again 
reflected,  forming  the  image 
/2.  The  position  of  this 
image  is  the  same  as  it 
would  be  if  /i  were  the 
actual  source  of  the  light.- 
(Why?)  A  part  of  this 
light  again  falls  upon  AB, 
forming  the  image  73,  just 
as  if  /2  were  its  source. 
With  the  given  angle  be- 
tween the  mirrors,  /a  is  the 
last  image  of  the  series;  for 
/3,  which  is  now  the  center 
of  the  reflected  waves,  lies 
behind  the  plane  of  the 
mirror  CD,  and  no  light 
from  this  point  or  from  its  direction  falls  upon  CD.  The  figure 
shows  the  path  of  the  light  by  which  /2  is  seen  from  the  position  E. 
There  is  also  a  second  series  of  three  images,  formed  by  reflection 
first  from  CD,  then  from  AB,  then  from  CD  again.  This  series  is 
not  shown  in  the  figure. 

The  smaller  the  angle  between  the  mirrors  the  greater  is  the  num- 
ber of  images  (Fig.  269).  When  the'  mirrors  are  parallel  the  series 
is  indefinite,  being  limited  only  by  the  gradually  failing  intensity  of 
the  light. 

PROBLEMS 

1.  What  is  the  length  of  the  shortest  plane  mirror  in  which  a  man  6  ft. 
tall  can  see  his  full-length  image? 

2.  If  a  person  stands  erect  before  a  plane  mirror  inclined  forward  at  an 
angle  of  30°,  at  what  angle  to  the  vertical  is  his  image? 

3.  (a)  In  what  respects  does  a  pinhole  image  differ  from  an  image  formed 
by  a  plane  mirror?     (b)  What  purpose  is  served  by  the  screen  upon  which 
a  pinhole  image  is  caught?      (c)  Can  an  image  formed  by  a  plane  mirror 
be  caught  upon  a  screen? 

4.  Why  are  stars  invisible  by  day? 

5.  Copy  Fig.  268  and  locate  the  other  series  of  images.     Construct  the 
path  of  light  to  the  eye  for  the  third  image  of  either  series. 


REFLECTION  OF  LIGHT  379 

6.  Draw  a  figure  of  mirrors  at  an  angle  of  90°,  locate  the  images  of  a 
point  object,  and  construct  the  path  of  light  to  the  eye  for  each  image. 

7.  Make  a  similar  diagram  with  the  mirrors  at  an  angle  of  60°,  and  the 
object  at  unequal  distances  from  the  mirrors.     Prove  that  a  point  object 
and  its  images  lie  on  the  circumference  of  a  circle. 

8.  If  a  kaleidoscope  is  provided,  examine  it  and  account  for  the  geo- 
metrical patterns  observed  in  it. 

9.  Show  by  means  of  a  diagram  that  in  the  case  of  parallel  mirrors 
a  point  and  its  images  lie  on  a  straight  line. 

313.   Reflection  from  a  Concave  Spherical  Mirror.  —  The 

reflecting  surface  of  a  concave  spherical  mirror  is  a  portion 
(usually  a  very  small  portion)  of  a  spherical  surface,  the 


TK 

FIG.  270.  —  Formation  of  a  Real  Image  by  a  Concave  Mirror. 

reflection  from  which  takes  place  on  the  inner  or  concave 
side.  Such  a  mirror  is  represented  in  a  section  diagram  by 
an  arc  of  a  circle,  as  MA7"  (Fig.  270).  The  center  of  curva- 
ture of  the  mirror,  C,  is  the  center  of  the  sphere  of  which 
the  mirror  is  a  part.  The  radius  of  this  sphere  is  called 
the  radius  of  curvature  of  the  mirror.  The  straight  line 
AC,  which  passes  through  the  center  of  curvature  and  the 
center  of  the  reflecting  surface,  is  called  the  principal  axis 
of  the  mirror;  any  other  line  passing  through  the  center 
of  curvature  to  the  mirror  is  a  secondary  axis. 

When  a  point  source  of  light,  S  (Fig.  270),  is  beyond 
a  certain  distance  from  a  concave  mirror,  the  divergent 
cone  of  incident  light,  MSN,  is  reflected  as  a  converging 


380  LIGHT 

cone,  MIN.  The  reflected  waves  are  concave,  with  /, 
the  vertex  of  the  cone,  as  their  center.  All  the  reflected 
light  from  the  point  5  comes  to  the  point  /,  where  it 
is  said  to  be  brought  to  a  focus.  Beyond  /  the  waves  are 
convex,  with  7  as  their  center,  just  as  if  this  point  were 
their  original  source.  The  reflected  rays  continue  in 
straight  lines  through  7,  where  they  all  meet  and  cross. 

When  the  reflected  light  falls  upon  a  screen,  it 
illuminates  .a  circular  area,  which  is  the  cross-section  of 
the  cone  at  that  place.  As  the  screen  is  brought  nearer 
to  7,  the  illuminated  area  becomes  smaller  and  brighter; 
at  7  it  is  a  bright  point,  and  the  light  is  then  said  to  be 
focused  upon  the  screen.  This  point  of  light  is  the  real 
image  of  the  point  source  S.  The  image  is  real  in  the 
sense  that  it  is  formed  by  light  which  actually  travels  to 
it  and  from  it;  and  it  exists  at  7  whether  caught  upon  a 
screen  or  not.  It  can  be  viewed  without  the  aid  of  a 
screen  from  any  position  within  the  cone  KIL,  and 
appears  to  be  out  in  space  where  it  really  is,  when  the 
eyes  are  directed  toward  it  (not  toward  the  more  distant 
mirror). 

If  the  point  source  is  at  7  its  image  will  be  at  5;  for 
the  path  of  any  incident  and  reflected  ray  from  S  to  7  is 
also  the  path  of  a  ray  from  7  to  S.  Any  two  points 
which,  like  S  and  7,  are  so  situated  with  respect  to  a 
concave  mirror  that  light  radiating  from  either  converges 
to  the  other  are  called  conjugate  foci. 

314.  Conjugate  Foci  on  the  Principal  Axis.  The  Prin- 
cipal Focus.  —  The  relative  positions  of  a  point  source  and 
its  image  can  best  be  accounted  for  and  most  easily  remem- 
bered by  considering  the  angles  of  incidence  and  reflec- 
tion of  the  rays.  These  angles  are  equal  for  any  ray,  as 


REFLECTION  OF  LIGHT  381 

is  always  the  case  in  regular  reflection,  whether  from  plane 
or  curved  surfaces.  The  perpendicular  at  any  point  of  a 
concave  mirror  is  always  a  radius  of  the  spherical  surface, 
and  passes  through  the  center  of  curvature,  as  M C  and  NC 
(Fig.  271).  Hence  the  angle  of  incidence  of  the  ray  SM  is 
SMC,  and  the  angle  of  reflection  is  CMG. 

The  cone  of  light  MSN  from  a  point  source,  5,  on  the 
principal  axis  and  beyond  the  center  of  curvature,  is  reflected 


FIG.  271.  —  Conjugate  Foci  on  the  Principal  Axis. 

as  a  converging  cone  MIN,  whose  vertex  /,  is  also  on  the 
principal  axis  and  on  the  opposite  side  of  the  center  of 
curvature.  /  is  the  focus  of  the  reflected  light  and  is  the 
image  of  S.  Its  exact  position  is  determined  in  the  figure 
by  constructing  any  two  incident  and  reflected  rays,  as 
SMG  and  SNE.  Any  two  will  serve,  since  all  intersect 
at  the  same  point.  We  might  regard  SA  as  one  of  the 
two,  for  it  is  incident  along  the  radius  CA  and  is  reflected 
back  along  the  same  path,  as  is  always  the  case  with  per- 
pendicular incidence. 

If  the  source  S  is  moved  away  from  the  center  of 
curvature  along  the  principal  axis,  the  angle  of  incidence 
i  increases;  and  since  the  angle  of  reflection  r  is  always 
equal  to  i,  it  is  evident  from  the  figure  that  the  focus  / 
moves  away  from  the  center  of  curvature  toward  the 
mirror.  When  5  is  moved  to  a  relatively  great  distance 
(100  times  the  radius  of  curvature  or  more),  the  incident 
rays  are  all  sensibly  parallel  to  the  principal  axis,  and 
their  focus  is  called  the  principal  focus  of  the  mirror. 


382  LIGHT 

This  can  be  shown  in  a  striking  manner  with  a  broad  beam  of 
sunlight,  falling  upon  a  concave  mirror  in  a  darkened  room.  The 
reflected  light  converges  to  a  small  spot  of  intense  brightness,  which 

lies  to  one  side  of  the  prin- 
cipal axis  when  the  mirror  is 
oblique  to  the  incident  beam 
(Fig.  272).  As  the  mirror 
is  turned  round  until  its  prin- 
cipal axis  is  parallel  to  the 
incident  rays,  the  focus  F' 
moves  with  it  to  the  point  F 
on  the  principal  axis.  This 

point  is  the  principal  focus. 
FIG.  272.  —  Focus  of* an  Oblique  Beam.        o*  •  ^i  i  •          i 

Strictly    speaking,    however, 

the  sun  is  only  approximately  equivalent  to  a  point  source,  since 
rays  from  opposite  sides  of  it  form  an  angle  of  half  a  degree  at  the 
distance  of  the  earth.  Hence  the  reflected  light  focuses  as  a  small 
round  spot,  which  is  an  image  of  the  sun  and  is  larger  than  a  point. 

The  principal  focus  lies  on  the  principal  axis  midway  be- 
tween the  mirror  and  the  center  of  curvature.  Its  distance 
from  the  mirror  is  called  the  principal  focal  distance  or  the 
focal  length  of  the  mirror,  and  is  equal  to  half  the  radius  of 
curvature.  This  is  the  least  possible  distance  of  a  real 
image. 

The  position  of  the  principal  focus  can  be  determined  by  geometry 
as  follows:  Let  BM  (Fig.  273)  be  any  incident  ray  parallel  to  the 
principal  axis,  F  the  point  where 
the  reflected  ray  cuts  the  axis, 
and  C  the  center  of  curvature. 
Angles  i  and  e  are  equal  (alter- 
nate-interior  angles  of  parallel 
lines),  and  angles  i  and  r  are  ^FOCUS.^ 

equal  (law  of  reflection).    Hence 

angles  r  and  e  are  equal,  the  triangle  MFC  is  isosceles,  and  MF 
=  FC.  If  MA  is  not  more  than  a  tenth  of  the  radius  "of  curva- 
ture (which  is  necessarily  the  case  if  the  mirror  is  to  give  distinct 


REFLECTION  OF  LIGHT  383 

images),  AF  is  very  nearly  equal  to  MF,  and  hence  also  to  FC;  i.e. 
AF  =  PC,  approximately.  For  points  of  incidence  at  and  immedi- 
ately about  A,  the  equality  is  exact;  and  this  determines  the  true 
position  of  the  principal  focus. 

Since  a  point  source  and  its  real  image  have  interchange- 
able positions,  the  above  discussion  may  be  summarized 
and  extended  as  follows:  The  image  of  a  point  source  at 
any  relatively  great  distance  on  the  principal  axis  is  at 
the  principal  focus.  As  the  source  moves  up  along  the  prin- 
cipal axis  to  the  center  of  curvature,  its  image  moves  from 
the  principal  focus  to  the  center  of  curvature,  where  source 
and  image  coincide.  (Why?)  As  the  source  moves  up 
from  the  center  of  curvature  to  the  principal  focus,  its  image 
moves  away  from  the  center  of  curvature  to  an  indefinite 
distance,  the  reflected  rays  being  then  parallel  to  the  prin- 
cipal axis.  As  the  source  moves  up  from  the  principal 
focus  to  the  mirror,  the  reflected  rays  become  more  and  more 
divergent,  and  the  image  is  virtual,  as  in  a  plane  mirror 
(Art.  316). 

315.  Conjugate  Foci  on  Secondary  Axes.  Real  Image 
of  a  Body  Object.  —  A  point  source  A  (Fig.  274  a,  b,  c,  and 
d)  and  its  image  A'  always  lie  on  the  same  axis,  i.e.  the 
same  straight  line  through  the  center  of  curvature  of  the 
mirror;  and  the  source  and  its  image  have  the  same  rela- 
tive positions  on  a  secondary  axis  as  on  the  principal  axis. 

In  locating  the  image  of  a  point  source  in  a  diagram,  it 
is  sufficient  to  find  the  point  of  intersection  of  any  two 
reflected  rays  from  the  given  point.  When  the  point  is 
not  on  the  principal  axis,  the  measurement  of  angles  can 
be  avoided  by  choosing  any  two  of  the  following  rays: 
(i)  The  incident  ray  passing  through  (or  from  the  direction 
of)  the  center  of  curvature,  which  is  reflected  back  along 


384  LIGHT 

the  same  path;  (2)  the  incident  ray  parallel  to  the  princi- 
pal axis,   which  is   reflected  through  the  principal  focus; 


FIG.  274.  —  Do,  Distance  of  Object;  Di,  Distance  of 
Image;  /,  Focal  Length  of  Mirror. 

(3)  the  incident  ray  passing  through  the  principal  focus, 
which  is  reflected  parallel  to  the  principal  axis. 

This  method  of  construction  serves  for  both  real  and  virtual  images, 
and  is  followed  in  Figs.  274,  275,  and  277.  In  constructing  the 
image  of  a  body  object  (usually  represented  by  an  arrow),  we  locate 
the  image  of  top  and  bottom  by  the  above  method.  The  line  con- 
necting these  points  represents  the  size  and  position  of  the  image  in 
true  proportion.  In  Fig.  274  a,  b,  and  d,  this  construction  is 
shown  for  the  image  of  A  only,  to  avoid  confusion. 


REFLECTION  OF  LIGHT  385 

Real  images  formed  by  concave  mirrors  are  always, 
inverted,  since  each  point  of  the  object  and  the  image 
of  that  point  are  on  the  same  axis  and  on  opposite 
sides  of  the  center  of  curvature,  and  all  axes  cross  at  this 
center. 

From  the  similar  triangles  ACB  and  A'CB'  of  Fig.  275, 
it  follows  that  the  size  (length)  of  the  image  is  to  the  size 


(N/ 


FIG.  275. 


of  the  object  as  the  distance  of  the  image  from  the  center 
of  curvature  is  to  the  distance  of  the  object  from  this  cen- 
ter. It  can  also  be  shown  that  these  distances  are  always 
in  the  same  ratio  as  the  distances  of  image  and  object 
from  the  mirror.  Hence  the  image  is  always  larger  than 
the  object  when  it  is  at  the  greater  distance  from  the 
mirror,  and  smaller  than  the  object  when  it  is  at  the  less 
distance. 


316.  Virtual  Images  by  Concave  Mirrors.  —  When  light 
from  a  point  source  at  the  principal  focus  is  j^ected  by 
a  concave  mirror,  the  reflected  waves  are  planeM«he  rays 
parallel.  The  light  does  not  come  to  a  focus^RT  no  di 
tinct  image  is  formed.  When  the  source  is  at  less 
the  focal  distance,  the  reflected  waves  are  convex  and  the 
rays  divergent  (Fig.  276),  but  less  so  than  when  the  reflec- 
tion is  from  a  plane  surface.  The  curvature  of  the  mirror 
has  the  effect  of  decreasing  the  curvature  of  the  reflected 
waves.  The  center  of  the  reflected  waves  is  their  virtual 


386 


LIGHT 


FIG.  276.  —  Formation  of  a  Virtual 
Image  by  a  Concave  Mirror. 


focus,  and  the  virtual  image,  i,  of  the  sources  is  at  this 
point. 

A  point  source  and  its  virtual  image  are  on  the  same  axis, 

as  A  and  A'  (Fig.  276).  The 
image  can  be  located  in  a  dia- 
gram by  producing  backward 
any  two  lines  representing 
rays  of  reflected  light.  Their 
point  of  intersection  is  the 
position  of  the  image.  By 
following  the  method  of  con- 
struction described  for  real  images  the  necessity  of  measur- 
ing angles  is  avoided  (see  Fig.  277).  As  shown  in  the 
figures,  the  distance  of  a  virtual  -image  is  always  greater 
than  that  of  the 
object.  (Why?) 
The  virtual  image 
of  a  body  object  is 
erect  and  magnified 
(Fig.  277).  As  the 
object  approaches  the  mirror,  the  image  also  approaches 
it  and  grows  smaller. 

317.  Spherical  aberration.  Parabolic  Mirrors.  —  As  we  have 
previously  noted,  the  reflecting  surface  of  a  concave  mirror  is  usually 
only  a  very  small  portion  of  a  spherical  surface.  This  must  be  so 
if  the  mirror  is  to  form  distinct  images.  As  a  rule,  the  angle  MCN 
(Fig.  275)  is  less  than  10°.  This  angle  at  the  center  of  curvature, 
formed  by  radii  extending  to  opposite  sides  of  the  mirror,  is  called 
its  angular  aperture.  When  the  aperture  is  large  the  images  are 
blurred;  for  only  a  part  of  the  reflected  light  from  each  point  of  the 
object  is  brought  to  the  corresponding  focus.  The  remainder  is 
more  or  less  widely  scattered.  This  effect  is  known  as  spherical 
aberration  (aberration  =  a  wandering  away).  It  is  illustrated  in 
Fig.  278  for  a  beam  of  light  parallel  to  the  principal  axis.  Only  the 


REFLECTION  OF  LIGHT 


387 


M 


central  portion  of  the  beam  is  focused  at  F;  and  the  marginal  rays 
"  wander  "  far  from  this  point. 

There  is  no  curved  reflecting  surface 
of  any  shape  that  will  accurately  focus 
all  reflected  light  from  a  point  source  in 
any  and  all  positions;  but  a  parabolic 
mirror  (Fig.  279)  does  accurately  focus  a 
beam  of  light  from  a  distant  point;  and 
conversely,  light  radiating  from  its  focus 
is  reflected  as  a  beam,  however  large  the 
aperture  of  the  mirror. 

Parabolic  mirrors  have  two  impor- 
tant uses:  (i)  in  headlights,  search- 
lights (Fig.  280),  etc.,  to  reflect  a 
strong  beam  in  a  definite  direction; 
(2)  in  reflecting  telescopes,  to  gather 
and  focus  the  light  of  heavenly  bodies. 
The  largest  telescopes  ever  constructed 
have  been  reflectors,  the  greatest  having 
a  mirror  6  ft.  in  diameter,  with  a  focal  length  of  60  ft.  Such 


J 

S^               c 

/KK* 

7/60°\ 

/ 

I      \ 

^•/v 

\          , 

f\\ 

\ 

/XXj 

\ 

/CNo^ 

/    /  '      \ 

22S 

zzz^  \ 

/^ji~ 

S^*^^"""""       \    ^ 

I^EfE 

^TY^^     / 

^"^ 

\  -HN^          / 

33 

/        < 

\A(i 

/ 

WV  A 

/ 

V 

/ 

\ 

\      / 

\\          a/ 

>X/60° 

^^.^^^ 

M' 

FIG.  278.  —  Spherical    Aberra- 
tion. 

FIG.  279.  —  Parabolic  Mirror. 


FIG.  280.  —  Search-Light  of  the  United 
States  Battleship  "Connecticut." 
Its  mirror  is  5  ft.  in  diameter. 


mirrors  have  very  little  curvature,  but  they  must  be  exceedingly 
accurate. 

318.   The  Convex  Spherical  Mirror.  —  By  reflection  from  a  convex 


388 


LIGHT 


mirror    the    plane   waves   of    a   beam    become    convex,    and    the 
beam  is  reflected  as  a  diverging  cone   (Fig.  281).     The  center  of 

the  reflected  waves  is  their 
virtual  focus.  The  convex 
waves  from  a  near  point 
source  are  made  still  more 
convex  by  such  reflection; 
hence  the  image  is  virtual 
and  nearer  the  mirror  than 
the  source  (Fig.  282). 

A  convex  mirror  always 
produces  or  increases  diver- 
gence   of     the    light,    and 
hence    forms    only    virtual 
images.      The    images    are 
and    smaller    than    the    object.    The 
midway   between   the   mirror   and 


FIG.  281.  —  Reflection  of    Plane  Waves 
a  Convex  Mirror. 


from 


282), 


s 


FIG.  282.  —  Formation  of  an  Image  by  a 
Convex  Mirror. 


always  erect  (Fig. 
image  of  a  distant  object 
its  center  of  curvature,  at 
the  principal  focal  distance, 
and  it  is  very  small.  As  the 
object  approaches  the  mir- 
ror, its  image  also  approaches 
the  mirror  and  grows  larger. 
The  definite  geometrical 
relations  between  the  radius 
of  a  convex  mirror,  its  prin- 
cipal focus  and  focal  length, 
and  the  relative  size  and  distance  of  object  and  image  can  all  be 
readily  established  on  the  same  principles  and  by  the  same  methods 
as  for  the  concave  mirror.  It  should  not  be  difficult  for  the  pupil 
to  solve  any  of  these  problems  in  which  he  may  chance  to  be 
interested. 

PROBLEMS 

1.  (a)  Account  for  the  difference  in  the  brilliance  and  distinctness  of  a 
pinhole  image  and  a  real  image  formed  by  a  concave  mirror.     (6)  A  pinhole 
image  is  an  imperfect  real  image.     Why  real?    Why  imperfect? 

2.  A  pinhole  image  can  be  seen  only  when  caught  upon  a  screen.    Why 
can  it  not  be  seen  in  the  air  like  a  real  image  formed  by  a  concave  mirror? 

3.  (a)  Prove  that  the  divergence  of  a  cone  of  light  is  not  changed  by 


REFRACTION  OF  LIGHT  389 

reflection  from  a  plane  mirror;    (6)  that  it  is  always  increased  by  reflection 
from  a  convex  mirror. 

4.  Why  do  plane  and  convex  mirrors  form  only  virtual  images? 

5.  What  are  the  essential  characteristics  of  a  virtual  image?   of  a  real 
image? 

IV.  REFRACTION  OF  LIGHT 

319.  Refraction.  —  When  light  falls  upon  the  surface 
of  still  water,  part  of  it  is  regularly  reflected,  forming  an 
image  of  the  source  as  in  a  plane  mirror.  The  remainder 
of  the  light  passes  into  the  water,  and  it  is  with  this  part 
that  we  are  now  concerned.  The  path  of  a  beam  of  light 
in  water  is  plainly  visible  in  a  darkened  room,  if  the  water 
is  clouded  with  a  small  quantity  of  an  alcoholic  solution  of 
mastic  or  a  little  milk.  A  rectangular  vessel  should  be 
used,  to  afford  a  view  of  the  water  through  a  flat  surface. 
It  will  then  be  observed  that,  when  a  beam  passes  obliquely 

into  the  water,  its  direction 
is  changed  at  the  surface, 
as  shown  in  Fig.  283.  The 
direction  of  the  bending  is 
toward  the  perpendicular  to 
the  surface  at  the  point  of 
incidence.  The  amount  of 
bending  is  less  when  the  in- 

FIG.  283. ^-Refraction.  cident  beam  is  more  nearly 

perpendicular  to  the  sur- 
face; and  when  the  incidence  is  exactly  perpendicular,  the 
path  is  one  straight  line. 

A  similar  abrupt  change  of  direction  takes  place  when 
light  passes  obliquely  from  almost  any  transparent  medium 
into  another,  and  the  phenomenon  is  called  refraction 
(breaking).  The  direction  of  bending  is  stated  with  ref- 
erence to  the  perpendicular  to  the  surface  at  the  point  of 


390  • 


LIGHT 


FIG.  284. 


incidence.  On  passing  obliquely  from  air  into  water, 
light  is  refracted  toward  the  perpendicular  MN  (Fig.  284). 
The  angle  i  between  this  perpendicular  and  the  incident 
ray  is  the  angle  of  incidence;  the  angle  r  between  the  per- 
pendicular and  the  refracted  ray  is  the  angle  of  refraction. 
The  change  of  direction  of  a  ray,  due  to  refraction,  is  called 
its  deviation  (angle  FOE  in  the  figure). 

The  path  of  light  is  reversible  in  refraction  as  in  reflec- 
tion. For  example,  if  the  ray  EO  (Fig.  284)  is  refracted 
in  the  direction  OH  on  passing  from 
air  into  water,  then  a  ray  passing  from 
water  into  air  and  incident  along  HO 
will  be  refracted  in  the  direction  OE. 
Hence  a  ray  passing  obliquely  from 
water  into  air  is  refracted  away  from 
the  perpendicular.  This  can  readily  be 
shown  by  placing  a  mirror  in  the  bottom  of  the  water 
tank  in  the  above  experiment,  to  reflect  the  beam  upward 
through  the  water  to  the  surface  again. 

320.  The  Cause  of  Refraction.  —  When  we  look  into  a 
vessel  of  water,  the  bottom  of  the  vessel  appears  to  be  raised 
above  its  true  position;  and  any  object  similarly  viewed 
appears  to  be  at  a  less  depth  than  it  really  is,  the  ratio  of 
apparent  to  real  depth  being  approximately  f  when  the 
line  of  sight  is  vertical  or  nearly  so.  It  is  evident,  there- 
fore, that  the  waves  of  light  from  any  point  under  water, 
as  5  (Fig.  285),  must  change  in  shape  as  they  pass  out 
into  the  air;  for  their  center  /,  as  they  travel  through  the 
air  to  the  eye,  is  the  apparent  position  of  the  point,  and  their 
center  while  they  are  still  in  the  water  is  its  true  position  S. 
Further,  since  a  point  under  water  appears  to  be  above 
its  true  position,  it  is  evident,  as  shown  in  the  figure,  that 


REFRACTION  OF  LIGHT 


391 


the  curvature  of  the  waves  must  increase  as  they  pass  out 
into  the  air.  This  means  that  a  wave  must  travel  the  dis- 
tance BB'  in  air  while  it  travels  the  less  distance  A  A'  in 
water;  hence  the  velocity  of  light  in  air  must  be  greater 
•than  it  is  in  water,  the  ratio  of  the  velocities  being 
BE'  :AAf.  It  can  be  shown  that  this  ratio  is  equal  to 
the  ratio  of  the  real  depth  of  the  point  to  its  apparent 
depth  (SB: IB)  when  the  point  is  viewed  perpendic- 
ularly to  the  surface;  and  it  is  found  by  measurement 
that  this  ratio  is  approx- 
imately f.  Hence  the 
velocity  of  light  in  air  is 
f  as  great  as  it  is  in 
water,  or  the  velocity 
in  water  is  f  as  great  as 
in  air.  This  agrees  with 
the  result  obtained  by 
direct  measurement  of 
the  velocity  in  water, 
an  experiment  first  per- 
formed by  the  French 
physicist,  Foucault,  in  1850.  Foucault's  experiment,  it  may 
be  remarked,  was  the  final  test  between  the  two  theories  of 
light;  for  the  emission  theory  involved  the  assumption  that 
light  travels  faster  in  water  and  other  refractive  media 
than  it  does  in  air. 

The  figure  explains  the  refraction  of  light  from  the  per- 
pendicular in  passing  obliquely  from  water  into  air.  It 
shows  further  that  rays  perpendicular  to  the  surface,  as 
SB,  pass  into  the  air  without  deviation.  Hence  an  object 
under  water  is  seen  in  its  true  direction  (but  not  at  its 
true  distance)  when  the  eyes  are  vertically  above  it.  It  is 
not  seen  in  its  true  position  from  any  point  of  view.  What 


FIG.  285.  —  Refraction  Caused  by  a 
Change  of  Speed. 


392 


LIGHT 


we  really  see  is  not  the  object,  but  its  virtual  image  formed 
by  refraction. 

The  refracted  waves  are  not  exactly  spherical  (Fig.  286).  If  they 
were,  the  apparent  position  of  any  object  under  water  would  not 

change  when  it  is  viewed 
more  and  more  obliquely; 
but  we  find  that  it  appears 
to  rise  toward  the  surface, 
as  shown  in  the  figure. 
From  the  position  E  it  is 
seen  at  7i,  from  F  it  appears 
to  be  at  72,  etc.;  hence  the 
curvature  of  such  a  wave 
must  be  slightly  greater 
near  its  margin.  One  in- 
teresting consequence  of  this 
FIG.  286.  —  Refraction  at  Different  Angles.  is  that  the  bed  of  a  lake 

which  is  really  of  uniform 

depth  has  the  appearance  of  a  deep  basin  surrounded  by  a  shoal, 
to  an  observer  looking  down  at  it  from  a  boat.  Standing  at 
the  margin  of  a  pool,  the  water  may  appear  to  grow  shallower 
at  some  distance  off  shore,  where  it  is  really  deeper,  —  a  deception 
which  has  doubtless  led  many  a  boy  beyond  his  depth. 

321.  Refraction  in  Different  Media.  —  Light  is  always 
refracted  on  passing  obliquely  from  one  transparent  me- 
dium into  another,  except  in  the  rare  case  when  its  velocity 
is  the  same  in  the  two  media.  The  greater  the  ratio  in 
which  the  velocities  differ  in  the  two  media  the  greater  will 
be  the  refraction  or  deviation  of  a  ray,  for  a  given  angle  of 
incidence.  For  example,  the  refraction  is  considerably 
greater  with  glass  and  air  than  it  is  with  water  and  air, 
the  velocity  of  light  in  glass  being  only  about  two  thirds 
as  great  as  it  is  in  air,  while  in  water  it  is  three  fourths 
as  great.  A  printed  page,  viewed  through  a  piece  of  thick 
glass  lying  upon  it,  appears  to  be  raised  one  third  the  thick- 


REFRACTION  OF  LIGHT 


393 


M 


Water 


Fig.  287. —  Refraction  from  Air  into 
Water. 


ness  of  the  glass,  owing  to  refraction  at  the  upper  surface. 
In  general,  the  less  the  velocity  of  light  in  any  medium  the 
greater  is  its  refractive  power. 

When  light  passes  from  one  medium  into  another  in 
which  its  velocity  is  less, 
as  from  air  into  water  or 
from  water  into  glass,  the 
curvature  of  the  waves  is 
diminished  and  the  rays 
are  bent  toward  the  per- 
pendicular (Fig.  287).  To 
an  observer  under  water 
an  object  in  the  air  appears 
to  be  at  a  greater  distance 
than  it  really  is. 

The  velocity  of  light  is 

less  in  ordinary  matter  of  any  kind  than  it  is  in  the  ether 
or  empty  space;  hence  the  refraction  is  toward  the  per- 
pendicular when  light  passes  into  any  substance  from  a 
vacuum,  and  from  the  perpendicular  when  it  passes  from 
any  substance  into  a  vacuum.  Air  is  practically  equiv- 
alent to  a  vacuum  in  all  phenomena  of  refraction  where 
the  second  medium  is  a  liquid  or  a  solid. 

322.  Laws  of  Refraction.  —  The  incident  and  refracted 
parts  of  a  ray  always  lie  on  opposite  sides  of  the  perpendic- 
ular at  the  point  of  incidence;  and  the  angles  of  incidence 
and  refraction  are  in  the  same  plane.  The  angles  are  un- 
equal, except  in  the  special  case  where  both  are  zero  (the 
incidence  being  perpendicular),  and  in  the  very  unusual 
case  where  .light  travels  with  equal  velocity  in  the  two 
media.  In  either  of  these  cases  there  is,  properly  speaking, 
no  refraction  at  all.  As  the  angle  of  incidence  increases 


394 


LIGHT 


the  angle  of  refraction  also  increases,  for  any  two  given 
media;  but  they  do  not  increase  in  the  same  ratio. 

The  exact  relation  between  these  angles  is  known  as  the  law  of 
refraction.  It  is  a  very  simple  relation,  but  troublesome  to  state, 
except  in  the  language  of 
trigonometry.  Let  AOB  and 
A'OB'  (Fig.  288)  be  the  path 
of  any  two  rays  from  one 
medium  into  another,  e.g. 


Air 


<\r 


Glass 

from  air  into  glass.  M  N  is 
the  surface  at  which  the  re- 
fraction takes  place;  KL  the 
perpendicular  at  the  point  of 
incidence.  Equal  distances 
OC,  OC,  OE,  and  OE'  are 
laid  off  along  the  incident 

and  refracted  rays.  This  is  conveniently  done  by  describing  a  circle 
about  the  point  of  incidence  as  a  center.  From  the  points  C,  C ',  E, 
and  Ef  perpendiculars  are  drawn  to  KL.  It  is  found  by  experiment 
that  the  ratio  of  CD  to  EF  is  equal  to  the  ratio  of  CD'  to  E'F; 

CD      C'D' 

This  ratio  differs  with  different  media, 


FIG.  288.  —  The  Law  of 
Refraction. 


I.e. 


a  constant. 


EF      E'F' 

but  is  constant  for  all  angles  of  incidence  with  the  same  two  media. 

In  trigonometry  the  ratio  of  a  side  of  a  right  triangle  to  the  hypothe- 
nuse  is  called  the  sine  of  the  angle  opposite  to  that  side;  i.e.  the  sine 

(~>T\  Tfjf 

of  angle  i  (abbreviated  to  sine  i )  is  -7^  and   sine  r  =  — •     Now 

CZ>  =  CD  +  CO  _  sine  I 
EF  = 


CO   "  EO 

Hence  the  law  of  refraction:   Whatever 


EF  +  EO       siner 
the  angle  of  incidence,  the  ratio  of  the  sine  of  the  angle  of  incidence  to 
the  sine  of  the  angle  of  refraction  is  constant,  for  the  same  two  media. 

It  can  be  shown  that  this  ratio  is  equal  to  the  ratio  of  the  velocities 
of  light  in  the  two  media. 

When  light  passes  from  a  vacuum  (or  air)  into  any  substance,  the 


REFRACTION  OF  LIGHT 


395 


Air 


.M 


ratio  of  the  sine  of  the  angle  of  incidence  to  the  sine  of  the  angle  of 

refraction  is  called  the  index  of  refraction  of  the  substance.      This 

ratio  measures  the  refractive  power 

of  the   substance.      It   is    constant 

for     all     angles     of    incidence,     in 

agreement  with  the  law  of  refraction; 

but    it    differs    very    slightly    with 

light    of      different     colors     (wave 

lengths),  as  we  shall  see  later.     The 

refractive    power    of    air    or    other 

gas,  although  very  slight,  increases 

with  its  density;  but  the  refractive 

powers  of  different  media  stand  in 

no  relation  whatever  to  their  relative 

densities  (see  table  following). 

The  index  of  refraction  of  a  sub- 


G/ 


Water 


FIG.  289.. —  Construction  for  the 
Refracted  Ray. 


stance  is  equal  to  the  ratio  of  the  velocity  of  light  in  a  vacuum 
(or  air,  approximately)  to  its  velocity  in  the  substance.  In  fact 
relative  velocity  is  at  the  root  of  the  whole  matter  of  refraction. 

INDICES  OF  REFRACTION 


SUBSTANCE 

INDEX  OF  FRACTION 

DENSITY  (G.  PER  CCM.) 

Diamond    

2.47  to  2.75 

3-5 

Carbon  bisulphide  

.63 

1.29 

Glass,  flint         .  .      . 

.^8  tO  1.7^ 

VOO  tO  V?2 

Glass,  crown 

.$2  tO  I."j6 

2.S 

Alcohol  

.36 

0.8o 

Ether,  sulphuric  

.36 

0.72 

Water  

•33 

I.  CO 

Air  ...                

.00029 

0.00129 

Vacuum    . 

.00 

323.  Construction,  for  the  Refracted  Ray.  —  Knowing  the  index 
of  refraction  of  a  substance,  we  can  determine  the  angle  of  refraction 
corresponding  to  any  given  angle  of  incidence  by  the  following 
method.  Let  EO  (Fig.  289)  be  a  ray  of  light  passing  from  air  into 
water  at  0.  Taking  f  as  the  index  of  refraction  of  water,  the 
construction  for  the  refracted  ray  is  as  follows:  Draw  KL  per- 


396  LIGHT 

pendicular  to  the  surface  M N  at  the  point  of  incidence;  and  from 
any  convenient  point,  C,  on  the  incident  ray  draw  CD  perpendicular 
to  KL.  Lay  off  on  M  N  a  distance  OF  equal  to  f  of  CD,  and 
construct  FB  perpendicular  to  MN  and  parallel  to  KL.  With  O 
as  a  center  and  a  radius  equal  to  OC,  describe  an  arc  cutting  FB  at 
G.  OG  is  the  refracted  ray.  (Prove  it.) 

To  construct  the  refracted  ray  for  light  passing  from  air  into  any 
substance,  take  OF  of  such  length  that  CD  :  OF  is  equal  to  the 
index  of  refraction  of  the  substance.  To  construct  the  refracted 
ray  for  light  passing  from  any  substance  into  air,  take  OF  of 
such  length  that  OF:  CD  is  equal  to  the  index  of  refraction  of 
the  substance. 

PROBLEMS 

1.  Draw  a  figure  showing  and  accounting  for  the  appearance  of  a  straight 
stick,  partly  immersed  in  water  in  an  oblique  position. 

2.  Stones  and  other  objects  lying  in  the  bed  of  a  brook  or  a  pond  appear 
to  dance  about  with  a  jerky,  irregular  motion  and  constantly  to  change  in 
shape,  as  small  waves  pass  over  the  surface  of  the  water.     Explain. 

3.  Look  through  common  window  glass  at  a  distant  building.     Why 
do  the  straight  lines  of  the  building  not  appear  straight?     Note  the  appear- 
ance of  these  lines  as  you  move  the  head  from  side  to  side.     Explain. 

4.  Upon  what  factors  or  conditions  does  the  deviation  of  a  ray  in  refrac- 
tion depend? 

NOTE.  — For  the  following  constructions  take  f  as  the  index  of  refraction 
of  crown  glass,  f  as  the  index  for  water,  and  f  (which  is  f  -f-  f )  as  the  relative 
index  of  refraction 'from  water  into  crown  glass. 

6.  Construct  the  path  of  a  ray  of  light  from  air  into  water  for  an  angle 
of  incidence  (a)  less  than  30°;  (b)  for  an  angle  between  40°  and  50°;  (c)  for 
an  angle  between  80°  and  90°. 

6.  Construct  the  path  of  a  ray  (a)  from  crown  glass  into  air;    (6)  from 
air  into  crown  glass. 

7.  Construct  the  path  of  a  ray  (a)  from  water  into  crown  glass;  (6)  from 
crown  glass  into  water. 

V.  REFRACTION  IN  SPECIAL  CASES.     TOTAL  REFLECTION 

324.  Refraction  through  a  Plate  having  Parallel  Sur- 
faces. —  Certain  cases  of  refraction  are  of  special  interest 


REFRACTION  IN  SPECIAL   CASES 


397 


and  importance,  owing  to  their  frequent  occurrence  or  their 
useful  applications.  These  cases  are  of  three  types,  repre- 
sented by  the  plate,  the  prism,  and  the  lens.  In  a  plate 
the  refracting  substance  is  bounded  by  two  parallel  plane 
surfaces,  in  a  prism  by  two  plane  surfaces  inclined  at  an 
angle  to  each  other,  and  in  a  lens  by  curved  surfaces,  which 
are  usually  spherical. 

When  a  ray  of  light  EOO'F  (Fig.  290)  passes  through  a 
transparent  body  having  parallel  plane  surfaces  AB  and 
CD,  the  angle  of  internal  inci- 
dence at  Of  is  equal  to  the 
angle  of  refraction  at  O;  hence 
the  angle  of  refraction  into 
the  air  is  equal  to  the  first 
angle  of  incidence.  The  emer- 
gent ray  O'F  is  therefore  par- 
allel to  the  incident  ray  EO} 
but  the  two  are  not  in  the 
same  straight  line.  The  re- 
sultant effect  of  the  two  refractions  is  a  displacement 
of  the  ray  to  one  side.  This  lateral  displacement  in- 
creases with  the  thickness  of  the  plate,  with  its  index 
of  refraction,  and  with  the  angle  of  incidence.  It  is  zero 
when  the  angle  of  incidence  is  zero,  and  is  small  for  all 
angles  of  incidence  with  thin  plates,  such  as  a  window- 
pane.  The  displacement  of  the  rays  when  an  object  is 
viewed  obliquely  through  a  glass  plate  causes  an  equal 
apparent  displacement  of  the  object.  Thus,  an  eye  at 
F  would  see  the  source  of  the  ray  at  some  point  on  the 
line  FH. 

A  beam  of  light  is  transmitted  as  a  beam  by  a  plate;  for 
the  incident  rays  are  parallel,  and  having  equal  angles  of 
incidence,  their  angles  of  refraction  are  also  equal,  and  the 


Air 


FIG.  290.  —  Refraction  through  a 
Parallel  Plate. 


Refraction  through  a  Prism. 


398  LIGHT 

refracted  rays  are  parallel.     This  means  that  plane  waves 

are  refracted  at  the  plane  surfaces  of  the  medium  without 

becoming  convex  or  concave  (see  figure). 

325.   Refraction  through  a  Prism.  —  When  light  passes 

through  a  refracting  medium  in  the  form  of  a  prism  (Fig. 

291),  its  direction  is 
necessarily  oblique 
to  one  of  the  sur- 
faces, and  it  gener- 
ally is  to  both.  In 
the  case  shown  in 
the  figure,  the  path 
of  the  ray  is  EFGH. 

The  first  refraction  is  toward  the  perpendicular  MN,  the 

second  from  the  perpendicular  M 'Nf ;  but  they  are  in  the 

same  direction  in  space,  and  the  resultant  deviation  is  their 

sum,  or  the  angle  KPH. 

The  angle  A  between  the  surfaces  through  which  the  light 

passes  is  called  the  refracting  angle  of  the  prism.     It  should 

be   remembered   that   the   resultant   deviation   is   always 

away  from  this  angle.     This  is  a  necessary  consequence  of 

the    retarding   effect   of   the 

medium  on  the  light  waves 

(Fig.   292).     That  part  of  a 

wave  which  is  farthest  from 

the  refracting   angle    travels 

the  greatest  distance  in  the 

prism,    and   consequently   is 

most  retarded   (the  velocity 

in  the  prism  being  less  than 

in  air).     This  is  true  for  all  angles  of  incidence,   hence 

the  waves  must  always  swing  round  from  the  refracting 

angle. 


FIG.  292.  —  Change  of  Speed  the 
Cause  of  Refraction. 


REFRACTION  IN  SPECIAL  CASES  399 

The  deviation  increases  with  the  refracting  angle  of  the  prism, 
and  with  its  index  of  refraction;  it  also  varies  with  the  angle  of  inci- 
dence, being  least  when  the  angle  of  incidence  is  such  that  the  angle 
of  emergence  is  equal  to  it.  The  deviation  varies  slightly  for  light 
of  different  colors,  producing  effects  which  are  considered  later. 

The  apparent  source  of  the  ray  GH  is  some  point  on  the  line  HL\ 
hence  an  object  viewed  through  a  prism  is  apparently  displaced  in 
the  direction  of  the  refracting  angle  of  the  prism. 

326.  Partial  and  Total  Reflection.  —  When  light  in  any 
medium  strikes  the  surface  of  another  transparent  medium 
which  has  either  a  greater  or  a  less  refractive  power  than  the 
first,  a  part  of  the  light  is  always  reflected,  and  under  cer- 
tain conditions  all  of  it  is.  In  the  first  case  the  reflection 
is  partial;  in  the  latter  it  is  called  total  reflection. 

The  partial  reflection  in  air  at  the  surface  of  water  has 
already  been  mentioned;  it  forms  the  images  we  see  in  still 
water.  Partial  reflection  in  water,  at  its  upper  surface, 
can  be  shown  in  a  darkened  room  with  a  beam  of  sunlight, 
reflected  upward  from  a  mirror  at  the  bottom  of  a  tank 
of  water,  as  in  a  former  experiment  in  refraction  (Fig.  283). 
The  greater  part  of  the  light  is  in  the  refracted  beam  EF; 
but  a  reflected  beam,  EG,  is  also  visible.  When  a  beam 
is  directed  obliquely  up  through  the  side  of  the  jar,  so  that 
the  angle  of  incidence  exceeds  48.5°,  all  the  light  is  reflected 
(Fig.  293).  This  is  a  case  of  total  reflection. 

That  partial  reflection  takes  place,  both  externally  and 
internally,  at  the  surface  of  glass  is  readily  shown  by  reflect- 
ing a  beam  of  sunlight  to  the  ceiling 
from  a  piece  of  colored  glass.  Two 
spots  of  light  will  appear  upon  the 
ceiling,  one  white,  the  other  having 
the  color  of  the  glass.  The  white  FlG>  293' 

light  is  reflected  at  the  front  surface  of  the  glass;  the  col- 
ored light  shows  by  its  color  that  it  has  traveled  through 


4oo 


LIGHT 


the  glass,  and  must  therefore  have  been  internally  reflected 
at  the  rear  surface. 

Internal  reflection,  both  partial  and  total,  can  be  admi- 
rably shown  by  means  of  a  glass  prism,  preferably  one  hav- 
ing an  angle  of  90°  and  two  angles  of  45°.  When  the  prism 
is  held  in  the  path  of  a  sunbeam  in  a  darkened  room,  in  the 
position  shown  in  Fig.  2940,  the  light  falling  upon  the  rear 
surface  at  B  is  partially  reflected  in  the  direction  BC  and 
partially  refracted  along  BD.  Both  beams  can  be  directed 
toward  the  ceiling,  where  their  relative  intensities  are  indi- 
cated by  the  relative 
brightness  of  the  two 
spots  of  light.  (The 
refracted  beam  pre- 
sents the  colors  of 
the  rainbow,  but  this 
effect  does  not  con- 

u-  u 

FIG.  294.  —  Partial  and  Total  Reflection.  cern    us    at   present.) 

As  the  prism  is  turned 

into  the  second  position  shown  in  the  figure,  the  inten- 
sity of  the  refracted  beam  decreases,  as  its  angle  of 
refraction  approaches  90°,  while  the  intensity  of  the  re- 
flected light  increases.  Finally,  in  the  position  shown,  the 
refracted  beam  disappears,  and  all  the  light  is  reflected 
in  the  direction  B'Cf.  It  should  be  noted  that  total  reflec- 
tion has  been  brought  about  by  increasing  the  angle  of 
internal  incidence  i' ,  which  is  now  about  45°,  and  that  the 
refracted  ray  BD  disappears  when  the  prism  is  turned 
beyond  the  point  where  the  angle  of  refraction  r\  is  90°. 
Total  reflection  can  take  place  'only  when  the  incident 
light  meets  a  less  refractive  medium  than  that  in  which 
it  is  traveling.  It  can  take  place  in  water  at  a  surface 
bounded  by  air,  but  not  at  a  surface  bounded  by  glass.  A 


REFRACTION  IN  SPECIAL   CASES 


401 


FIG.   295. 


further  necessary  condition  is  that  the  angle  of  incidence 

must  be  greater  than  that  for  which  the  angle  of  refraction 

is    90°.      The   angle   of    incidence    in 

the     more     refractive     medium     for 

which   the  angle   of   refraction  is   90° 

is    called    the    critical    angle.     When 

the  second  medium  is  not  mentioned, 

it  is  assumed  to  be  air.     The  critical 

angle  for  water  and  air  is  48.5°,   for  crown  glass   and 

air  it  is  about  41°,  for  flint  glass  and  air  38°,  for  diamond 

and  air  24°. 

When  a  face  of  a  prism  is  viewed  internally  at  such  an 

angle  that  the  eye  receives  light  from  it  by  total  reflec- 
tion, it  has  the  brilliant,  silvery  appear- 
ance of  a  perfect  mirror.  In  fact,  the 
most  perfect  mirrors  that  can  be  made 
are  total-reflecting  prisms,  and  on  this 
account  they  are  much  used  in  optical 
instruments.  For  example,  a  right  angled 
prism  at  the  eye  end  of  an  astronomical 
telescope  changes  the  direction  of  the 
light  by  90°  (Fig.  295),  and  enables  the 
observer  to  look  obliquely  downward  in 
viewing  the  heavenly  bodies,  thus  avoid- 
ing the  tiresome  position  that  must  be 

assumed    in    looking    upward    in    the    direction    of    the 

object. 

Total  reflection  is  usefully  applied  in  the  natural  and  artificial 
lighting  of  buildings.  Where  windows  of  a  store  or  office  face  a 
narrow  court,  the  amount  of  light  admitted  through  ordinary  window 
glass  is  often  insufficient.  In  such  cases  the  lighting  is  greatly  im- 
proved by  using  luxfer  prism  glass,  the  outer  side  of  which  is  formed 
into  angular  ridges  or  prisms,  running  horizontally.  In  a  vertical 


FIG.  296.  —  Luxfer 
Prism  Glass. 


402 


LIGHT 


cross-section  these  ridges  appear  like  saw-teeth   (Fig.  296).    The 
light  coming  from  a  nearly  vertical  direction  strikes  the  upper  surface 

of  the  prisms  less  obliquely  than 
it  would  upon  the  vertical  surface 
of  plane  glass.  This  diminishes 
the  loss  due  to  external  reflection. 
The  light  that  penetrates  the  glass 
is  refracted  and  internally  re- 
flected as  shown  in  the  figure, 
being  thus  directed  toward  the 
Extensive  Reflector.  walls  and  ceiling,  while  with  ordi- 


Intensive  Reflector.  Focusing  Reflector. 

FIG.  297.  —  Types  of  Holophane  Reflectors. 

nary  glass  it  would  fall  directly  on  the  floor  and  be  almost  wholly 
lost. 

Prismatic  reflectors,  shades,  and  globes  of  the  so-called  holophane 
type  are  very  effective  as  a  means  of  distributing  and  diffusing  arti- 
ficial light.  By  refraction  and  internal  reflection  the  light  is  directed 
downward,  and  more  or  less  concentrated,  according  to  the  shape  of 
the  reflector.  Three  types  of  distribution  are  shown  in  Fig.  297. 
Similar  results  are  obtained  with  holophane  globes,  the  character 
of  the  distribution  being  determined  by  the  shape  of  the  prisms. 

327.  Atmospheric  Refraction.  —  Although  the  refractive  power 
of  the  air  is  small,  it  is  responsible  for  certain  rather  curious  and 
interesting  phenomena.  Objects  seen  through  currents  of  heated 


REFRACTION  IN  SPECIAL   CASES  403 

air  rising  from  a  bonfire  or  a  hot  stove,  or  from  the  ground  on 
a  hot  summer  day,  seem  to  quiver  and  to  shift  about  with  a  slight, 
unsteady  motion.  Tips  anngaj^nceis  dii£_tQ  the  (jpnst.fl.ntly  rhan- 
gn  refra.rt.inn  of  Tjfre.  ig^t  as.  it  passes  through  the  unequally 
for  the  density  of  the  air  varies  with  its 
temperature,  and  its  refractive,  .power  varies  with  its  densjty.  The 
unsteady  condition  of  the  air  itself  can  be  seen  when  the  .light 
isJavorabJe. 

The  twinkling  of  the  stars  is  an  atmospheric  phenomenon.  The 
stars  themselves  are  fixed  and  shine  with  a  steady  light.  The  twinkling 
is  caused  by  the  changing  refraction  of  the  light  as  currents  of  air  of 
varying  density  cross  the  line  of  sight.  As  a  beam  of  light  passes 
through  successive  layers  of  air,  the  refraction  at  their  irregular 
boundaries  may  cause  either  a  slight  convergence  or  divergence  of 
the  rays.  Convergence  increases  the  intensity  of  the  beam,  diver- 
gence diminishes  it;  and  the  twinkling  is  largely  due  to  the  rapid 
alternation  of  these  effects.  Stars  near  the  horizon,  the  light  from 
which  traverses  a  greater  stretch  of  atmosphere,  twinkle  more  than 
those  overhead.  The  twinkling  also  differs  greatly  on  different  nights, 
according  to  the  steadiness  of  the  air. 

The  inconstant  and  irregular  refraction  to  which  the  twinkling 
of  the  stars  is  due  is  small  in  comparison  with  the  regular  atmospheric 
refraction,  due  to  the  increasing  density  of  the  atmosphere  from  its 
upper  limit  to  the  earth's  surface.  Light  traveling  obliquely  down- 
ward through  the  atmosphere  is  bent  continuously  toward  the  per- 
pendicular (Fig.  298).  The  total  deviation  varies  from  zero,  for 
heavenly  bodies  directly  overhead, 
to  a  little  more  than  half  a  degree 
at  the  horizon.  (It  is  greatly  ex- 
aggerated in  the  figure.)  Since  the 
angular  diameter  of  the  sun  at  the 
earth  is  about  half  a  degree,  the 

sun  is  really  just  below  the  hori-     FlG-  298-  —  Atmospheric  Refraction. 

i  i  (Much  exaggerated.) 

zon  when  it  appears  to  be  just 

above  it.  Thus,  on  account  of  atmospheric  refraction,  sunrise  occurs 
from  two  to  four  minutes  earlier  than  it  otherwise  would  (varying 
with  the  angle  that  the  sun's  path  makes  with  the  horizon),  and 
sunset  is  retarded  by  the  same  amount. 

The  mirage  is  a  most  interesting  optical  illusion,  due  to  atmos- 


404  LIGHT 

pheric  refraction.  It  is  most  frequently  observed  in  hot,  desert 
regions,  where  it  presents  the  appearance  .of  a  tranquil  lake  in  the 
distance,  in  which  the  traveler  sees  the  reflection  of  the  sky  and  the 
scattered  trees  or  other  objects  of  the  landscape.  But  no  water  is 


FIG.  299.  —  The  Mirage. 


there;  the  reality  is  th^  hot, 


o,f  the 


The  reflection 


of  the  light  takes  place  in  the  lower  layers  of  heated  air  near  the 
ground^    The  lowest  layers  are  the  hottest,  and,  having  expanded  the 


FIG.  300.  —  Looming. 

e  less_ dense  and  less  refractive  than  the  air  abov.e.  Hence 
a  ray  of  light,  ADE  (Fig.  299),  traveling  obliquely  downward  is 
refracted  from  the  perpendicular,  as  it  passes  through  successive  layers 


LENSES 


405 


of  air  near  the  ground;  and  if  its  course  is  nearly  horizontal,  it  will  fi- 
nally meet  a  layer.of  less  refractive  air  at  an  angle  of  inqjjence  greater 
anfl1pi  and  will  then  be  totally  reflected.  The  ray 


is  refracted  toward  the  perpendicular  as  it  returns  through  the  denser 
air  above.  By  this  refraction  and  total  reflection  images  are  formed 
like  those  seen  in  the  surface  of  still  water.  The  sky  and  other 
objects  are  also  seen,  at  the  same  time,  erect  and  in  their  true  posi- 
tion^ hy  light  that  rnmpfi  straight  to  the  eye;  hence  the  illusion  Js 


perfect. 

A  similar  phenomenon,  called  looming,  is  occasionally  seen  over 
the  sea  in  still,  hot  weather.  The  image  of  a  distant  ship  appears  in 
the  sky,  sometimes  inverted,  sometimes  upright.  In  such  cases  the 
total  reflection  takes  place  where  the  light  meets  an  upper  layer  of 
warm,  still  air  (Fig.  300). 


VI.  LENSES 

328.  Forms  of  Lenses.  —  A  lens  is  a  transparent  body 
bounded  by  two  curved  surfaces,  or  by  a  plane  and  a  curved 
surface.  Lenses  are  usually  made  of  glass,  and  their  curved 
surfaces  are  usually  spherical.  There  are  six  forms  of 
spherical  lenses,  sections  of  which  are  shown  in  Fig.  301. 


a 
VI 


L\ 

FIG.  301.  —  a,  b,  and  c,  are  Converging,  and  d,  e,  and/, 
Diverging   Lenses. 


The  first  three  are  of  the  type  known  as  convex  or  conver- 
ging lenses;  the  last  three  are  concave  or  diverging  lenses. 
Convex  lenses  are  all  thickest  at  the  middle,  concave  lenses 
thinnest  at  the  middle.  The  different  forms  of  lenses  are 
adapted  to  different  special  uses  in  optical  instruments; 


406 


LIGHT 


FIG.  3026. 


but  the  double  convex  lens,  a,  and  the  double  concave  lens, 
d,  are  typical  and  serve  for  experimental  work. 

329.   Effects  of  a  Convex  Lens  on  Light.  —  When  light 
passes  through  a  convex  lens,  the  central  part  of  each 

transmitted  wave  is 
most  retarded,  since 
it  passes  through 
the  greatest  thick- 
ness of  glass.  From 
the  center  out  to 
the  margin  of  the 
lens,  on  all  sides, 
the  retardation  of 
the  wave  grows  less 
as  the  length  of 
its  path  through 
the  glass  decreases. 
This  action  of  the 
lens  produces  the 
following  results: 

(i)  Plane  waves 
are  changed  to  concave  waves,  which  converge  to  a  real 
focus  (Fig.  3020). 

(2)  Convex  waves,  if  their  curvature  is  not  too  great, 
are  also  changed  to  concave  waves,  but  of  less  curvature 
than  in  the  first  case,  and  the  focus  is  at  a  greater  distance 
(Fig.  3026). 

(3)  Convex  waves  of  a  certain  degree  of  curvature  are 
refracted  as  plane  waves.     This  is  the  first  case  with  the 
direction  in  which  the  light  is  traveling  reversed. 

(4)  When  the  curvature  of  the  incident  waves  is  still 
greater,  the  refracted  waves  are  convex,  but  less  so  than 


FIG.  3o2c. 
FIG.  302.  —  Effects  of  a  Converging  Lens. 


LENSES  407 

the  incident  waves  (Fig.  302^).     In  this  case  the  focus  is 
virtual. 

These  effects  will  be  recognized  as  identical  with  those 
produced  by  a  concave  mirror.  A  convex  lens  forms  a 
real  or  a  virtual  image  of  the  source  of  light,  depending 
upon  the  converging  power  of  the  lens  and  the  distance 
of  the  source  from  it.  The  behavior  of  the  light  in 
forming  these  images  is  the  same  as  with  mirrors; 
but  the  action  of  the  lens  in  causing  this  behavior  is 
different  from  that  of  mirrors  and  presents  a  new  prob- 
lem. The  geometrical  relations  involved  are  more  simply 
presented  by  considering  the  rays  of  light  rather  than 
the  waves. 

330.  Conjugate  Foci  on  the  Principal  Axis.  The  Prin- 
cipal Focus.  —  The  straight  line,  XY  (Fig.  302),  through 
the  centers  of  curvature,  C  and  C',  of  the  spherical  surfaces 
of  a  convex  lens  is  called  its  principal  axis.  This  line  also 
passes  through  the  center  of  the  lens,  O.  A  ray  of  light 
incident  along  the  principal  axis  continues,  as  an  emer- 
gent ray,  along  that  axis;  for  it  meets  both  surfaces  of  the 
lens  perpendicularly  and  is  not  refracted.  Hence  when 
a  point  source  S  (Fig.  302  a,  6,  and  c)  is  on  the  principal 
axis,  its  image  /,  whether  real  or  virtual,  is  also  on  that 
axis.  In  Fig.  3020  the  point  source  is  at  a  relatively  great 
distance,  and  the  incident  rays  are  parallel  to  the  principal 
axis.  The  position  of  the  point  image  in  this  ease  is  called 
the  principal  focus,  and  its  distance  from  the  lens  is 
called  the  principal  focal  distance  or  the  focal  length  of 
the  lens.  This  case  is  approximately  shown  for  the  lens, 
as  it  is  for  the  concave  mirror,  when  a  sunbeam  is  inci- 
dent parallel  to  the  principal  axis;  for  the  light  converges 
to  a  small  round  spot  (the  image  of  the  sun)  at  the  princi- 


408 


LIGHT 


pal  focus.     (There  is  a  principal  focus  at  the  same  dis- 
tance on  each  side  of  a  lens.) 

A  point  source  and  its  real  image  formed  by  a  lens  are 
at  conjugate  foci :  light  radiating  from  either  point  converges 
to  a  focus  at  the  other.  When  the  point  source  is  at  a 
relatively  great  distance  on  the  principal  axis,  its  image 
is  at  the  principal  focus  (Fig.  3020).  As  the  source  moves 
toward  the  lens,  the  image  recedes  from  it.  (Why?) 
When  the  source  is  at  twice  the  focal  length,  the  image  is 

also  at  twice  the  focal  length. 
As  the  source  moves  up  to  the 
principal  focus,  its  image  re- 
cedes to  an  indefinite  distance. 
When  the  source  is  nearer 
than  the  principal  focus,  the 
refracted  waves  are  convex 
and  the  image  is  virtual  (Fig. 
302^).  If  the  distance  of  the 
source  is  only  very  slightly 

less  than  the  focal  length,  the  refracted  waves  are  very 
nearly  plane,  and  the  virtual  image  is  at  a  great  distance. 
As  the  source  moves  up  from  the  principal  focus  to  the 
lens,  its  virtual  image  moves  up  from  a  very  great  dis- 
tance to  the  lens;  but  it  is  always  at  a  greater  distance 
than  the  source.  (Why?)  The  real  image  is  always,  on 
the  opposite  side  of  the  lens  from  the  source,  and  the  vir- 
tual image  is  on  the  same  side. 

The  focal  length  of  a  lens  depends  jointly  upon  the  cur- 
vature of  its  surfaces  and  the  index  of  refraction  of  the 
glass.  Experiment  shows  (and  it  can  be  proved  mathe- 
matically) that,  when  the  index  of  refraction  is  1.5  and  the 
faces  of  the  lens  have  equal  curvature,  the  principal  focus 
is  at  the  center  of  curvature  of  either  face.  Since  this  is 


FIG.  303.  —  Thick  and  Thin 
Lenses. 


LENSES  409 

approximately  the  index  of  refraction  of  crown  glass,  it 
may  be  assumed  in  constructing  diagrams  that  the  prin- 
cipal focus  is  at  the  center  of  curvature  of  either  surface. 
The  converging  power  of  a  lens  increases  as  its  focal  length 
decreases.  In  popular  language  lenses  are  "stronger"  or 
"weaker"  according  to  their  greater  or  less  converging 
power.  We  can  roughly  estimate  the  focal  length  of  a  lens 
of  given  diameter  from  its  thickness  (Fig.  303). 

331.  Conjugate  Foci  on  Secondary  Axes.  Real  and 
Virtual  Images  of  a  Body  Object.  —  Any  straight  line 
through  the  center  of  a  lens,  other  than  the  principal  axis, 
is  called  a  secondary  axis,  as  A  A'  (Fig.  304).  A  ray  of 
light  incident  along  a 
secondary  axis  strikes 
the  surface  of  the  lens 
obliquely  and  is  re- 
fracted, but  on  emerging 
from  the  lens  it  is 
equally  refracted  in  the 

Opposite    direction,    just    F^   304--Ray   through   the    Center  of   a 

as     if    it    had    passed 

through  a  "plate."  The  lateral  displacement  of  the  ray  is 
slight,  especially  if  the  thickness  of  the  lens  is  only  a  few 
millimeters,  and  in  the  elementary  treatment  of  lenses  it  is 
disregarded.  A  point  and  its  image,  whether  real  or  virtual, 
are  therefore  on  the  same  axis  (Figs.  305  and  306). 

Since  all  axes  cross  at  the  center  of  the  lens,  real  images, 
being  on  the  opposite  side  of  the  lens  from  the  object,  are 
inverted,  and  virtual  images,  being  on  the  same  side  of 
the  lens,  are  erect. 

A  real  image  can  be  seen  by  focusing  it  on-  a  screen.  It 
is  also  directly  visible,  in  its  true  position  in  space,  when 


LIGHT 

the  eyes  of  the  observer  are  within  the  path  of  the 
light  diverging  from  it.  The  observer  must  look  toward 
the  lens,  but  at  the  image,  which  is  nearer  than  the  lens. 


A' 

FIG.  305.  —  Formation  of  Real  Image  by  a  Convex  Lens. 

A  virtual  image  is  seen  by  looking  through  the  lens. 
In  unscientific  language  it  is  termed  the  "magnified 
object." 

The  image  of  a  point  source  is  located  in  a  diagram  by 
constructing  the  path  of  two  refracted  rays  from  the  point. 
If  the  rays  are  convergent,  the  image  is  at  their  point  of 
intersection;  if  they  are  divergent,  it  is  at  their  apparent 
source.  In  the  latter  case  the  lines  representing  the 
refracted  rays  are  produced  backward  to  their  point  of  inter- 
section. The  necessity  of  measuring  angles  of  incidence  and 
refraction  can  be  avoided  by  choosing  any  two  of  the  follow- 
ing rays,  (i)  The  ray 
along  the  axis  which 
passes  through  the  point 
source.  This  continues 
in  the  same  straight 
line.  (2)  The  ray  paral- 
lel to  the  principal  axis. 

FIG.  306.  —  Formation  of  a  Virtual  Image  by    This  paSSCS  through  the 
Convex  Lens.  •    .      .      .   .  . 

principal  focus  after  re- 
fraction. (3)  The  incident  ray  passing  through  the 
principal  focus  on  the  same  side  as  the  object.  This 
is  refracted  parallel  to  the  principal  axis.  This  method 
of  construction  is  illustrated  by  Figs.  307,  308,  and  309. 


LENSES 


411 


332.   Relative  Size  and  Distance  of  Object  and  Image. 

—  The  relative  size  of  image  and  object  is  a  matter  of  the 
first  importance  in  the 
use  of  lenses  in  optical 
instruments.  The  fol- 
lowing relations  should 
therefore  be  carefully 

.          .  ,          ,  FIG.  307.  —  Image  of  a  Distant  Object. 

noted  and  remembered. 

From  the  similar  triangles  AOB  and  A' OB'  (Figs.  308 
and  309),  A'B'  :  AB  : :  D'O :  DO;  i.e.  the  size  (length)  of 
the  image  (real  or  virtual)  is  to  the  size  of  the  object  as  the 
distance  of  the  image  from  the  lens  is  to  the  distance  of 
the  object  from  the  lens. 

For  a  given  object  at  a  given  distance,  the  size  of  the  real 
image  increases  with  the  focal  length  of  the  lens;  since,  under 
these  conditions,  the  greater  the  focal  length  the  greater 
is  the  distance  of  the  image  from  the  lens.  (Illustrate 
with  two  diagrams,  constructed  for  lenses  of  unequal  focal 
length.)  An  important  special  case,  relating  to  the  use 
of  the  telescope,  is  that  of  a  distant  object  (Fig.  307).  In 
this  case  the  distance  of  the  image  is  the  focal  length  of 
the  lens,  and  the  size  of  the  image  is  proportional  to  the  focal 
length  of  the  lens.  (Draw  figures  to  illustrate.) 

When  a  lens  is  used  as  a  simple  microscope  or  a  "magni- 
fying glass"  in 

A  r  *  ^^-^^^          looking  at  small 

objects,  the  ob- 
ject and  the  lens 
are  so  adjusted 
that  the  distance 
of  the  magnified 

virtual  image  is  about  12  or  14  in.,  or  the  distance  at 
which  a  book  is  held  for  reading.     Hence  in  studying  the 


FIG.  308.  —  Real  Image. 


412  LIGHT 

effect  of  greater  or  less  focal  length  on  the  size  of  the 
virtual  image,  we  are  interested  only  in  the  case  where  the 
distance  of  the  image  is  the  same  with  the  different 
lenses.  With  this  adjustment,  the  shorter  the  focal  length 
of  the  lens  the  larger  is  the  virtual  image.  (Draw  figures 
to  illustrate.) 

An  important  general  fact,  then,  is  this:  Other  conditions 
remaining  the  same,  larger  real  images  are  formed  by  convex 
lenses  of  greater  focal  length,  larger  virtual  images  by  con- 
vex lenses  of  shorter  focal  length. 

333.  Relation  between  Conjugate  Focal  Distances  and 
the  Focal  Length  of  a  Convex  Lens.  —  From  the  similar 
triangles  AOD  and  A'OD'  (Fig.  308), 

AD-.A'D'  =  OD-.OD'. 
From  the  similar  triangles  EOF  and  A'D'F, 

EO'.A'D'  =  OF-.FD'. 
Since  AD  =  EO,  we  have  from  these  proportions 

OD:OD'  =  OF'.FD'. 

Let  OD  be  denoted  by  D0  (object  distance),  OD'  by  D{ 
(image  distance),  and  OF  by  f  (focal  length);  then  the  last 
proportion  becomes  — 


From  which  DJ  =  DQD{  -  DJ. 

Transposing,  DJ  +  DJ  =  D0D{. 

Dividing    by    D0DJ,   jr  +  —  =  7-    (Formula  for  real  images.) 

UQ  L)\  J 

By  means  of  this  formula  we  can  find  any  one  of  the 


LENSES 


413 


three  quantities,   D0,  Dv   and  /  when  the  other  two  are 
known. 

The  formula  for  virtual  images  is  derived  as  follows: 
From  the  similar  triangles  A'EA  and  A'FO  (Fig.  309), 

A'A-.A'O  =  AE-.OF. 
From  the  similar  triangles  AOD  and  A'OD', 

A'A-.A'O  =  D'D:DfO. 
Hence  D'D:  D'O  =  AE  :  OF. 

Representing  the  distances  of  object  and  image  and  the 
focal  length  by  the  letters  D0,  D{  and  /  respectively,  the 
last  proportion  becomes  — 

(A-  A>):A  =  A>:/. 
From  which  DJ  -  DJ  =  D0D{. 


Dividing  by  D0D-J,  j^  --  jr 


7-  (Formula  for  virtual  images.) 

J 


334.  The  Concave  or  Diverging  Lens.  —  When  light 
passes  through  a  concave  lens,  the  central  part  of  each 
transmitted  wave  is 
least  retarded,  since 
it  passes  through  the 
least  thickness  of 
glass.  From  the  cen- 
ter out  to  the  margin 
the  retardation  of  the  B 
wave  increases  with 
the  increasing  thickness  of  the  lens;  hence,  in  passing 
through  the  lens,  the  marginal  portion  of  a  wave  lags 
behind  its  center.  Plane  waves  are  thus  changed  to  con- 
vex waves  (Fig.  3100),  and  convex  waves  are  made  more 
convex  (Fig.  3106).  When  the  incident  light  is  a  beam 


FIG.  309.  —  Virtual  Image. 
i 


414 


LIGHT 


parallel  to  the  principal  axis,  the  center  of  the  refracted 

waves  is  a  point  on  the  principal  axis,  and  is  called  the 

principal  focus  of   the  lens.     It  is,  of   course,   a   virtual 

focus. 
Whatever  the  position  or  distance  of  a  point  source,  its 

image  formed  by  a  concave  lens  is  virtual.     It  is  on  the  same 

\  side  of  the  lens 
\  Ju  and  on  the 
same  axis  as 
the  source,  and 
is  at  a  less  dis- 
tance from  the 
lens.  As  the 
source  moves 
up  from  a  great 
distance  to  the 
lens,  the  image 
moves  up  from 
the  principal 

focal  distance  to  the  lens.     The  image  of  a  body  object 

is  erect  and  smaller  than  the  object  (Fig.  311),  like  the 

images  formed  by  convex  mirrors. 

Concave  lenses  are  used  in  combination  with  convex 

lenses  in  optical  instruments,  and  are  worn  as  eye-glasses 

to  correct  short  sight  (Art.  340). 

335.  Spherical  Aberration.  —  No  single  lens,  whatever 
its  shape,  brings  all  the  light  that  passes  through  it  from  a 
point  source  exactly  to  the  conjugate  focus.  There  is 
always  an  imperfection  of  focusing,  called  chromatic  (or 
color)  aberration;  and  with  spherical  lenses  there  is  an- 
other imperfection,  known  as  spherical  aberration.  We 
are  at  present  concerned  only  with  the  latter. 


FIG.  310.  —  Effects  of  a  Concave  Lens. 


LENSES 


415 


FIG.  311.  —  Image  by  a  Concave  Lens. 


The  outer  or  marginal  part  of  a  spherical  lens  refracts 
the  light  to  a  nearer  focus  than  the  central  part  of  the  lens 
(Fig.  312).,  This  causes 
a  blurring  of  the  image  A 
and  loss  of  detail,  which 
is  especially  marked  with 
lenses  of  short  focal 
length.  The  defect  can 
be  remedied  in  three 

ways,  (i)  By  using  an  opaque  diaphragm  with  a  small 
circular  opening,  which  admits  light  only  to  the  central 
part  of  the  lens.  This  device  is  often  resorted  to  where 
only  a  small  amount  of  light  is  needed,  as  in  small  and 
inexpensive  cameras.  (2)  By  decreasing  the  curvature  of 
the  surfaces  of  the  lens  near  the  edge.  This  is  a  method 
adopted  for  the  large  lenses  of  astronomical  telescopes. 
The  grinding  and  polishing  of  such  surfaces  must  be  done 
by  hand.  This  requires  exceptional  skill,  and  is  a  very 
slow  and  costly  process.  (3)  By  using  a  set  of  two  or 


FIG.  312.  —  Spherical  Aberration. 

more  lenses  to  do  the  work  of  one.  Such  sets  can  be 
constructed  so  as  to  correct  both  spherical  and  chromatic 
aberration,  and  are  regularly  used  in  first-class  cameras, 
microscopes,  etc. 

PROBLEMS 

1.   State  and  account  for  the  points  of  resemblance  and  of  difference 
between  the  real  image  formed  by  a  lens  and  a  pinhole  image. 


416 


LIGHT 


2.  Show  from  the  lens  formula  that  (i)  when  D0  is  very  great,  D{  =  /; 
(2)  when  DO  =  f,  Di  is  very  great;   (3)  when  D0  =  2/,  D{  =  2/5   (4)  that 
DI  increases  as  D0  decreases,  and  vice  versa,  for  real  images;   and  (5)  that 
Di  decreases  as  D0  decreases,  for  virtual  images. 

3.  An  object  2  cm.  long  is  at  a  distance  of  50  cm.  from  a  lens  whose 
focal  length  is  15  cm.     Find  the  distance  of  the  image  and  its  length. 

4.  An  object  i  cm.  long  is  at  a  distance  of  1.7  cm.  from  a  lens  whose 
focal  length  is  2  cm.     Find  the  distance  of  the  image  and  its  length. 


VII.  THE  EYE 

336.  The  Eye  as  an  Optical  Instrument.  —  The  fore- 
going optical  principles  are  beautifully  exemplified  in  the 
structure  and  action  of  the  eye. 

The  human  eye  (Fig.  313)  is  a  nearly  spherical  ball  some- 
what less  than  an  inch  in  diameter.  Its  thick  outer  coat 
or  wall  is  opaque  and  white,  except  the  part  in  front,  which 
is  transparent.  This  part  is  called  the  cornea.  Behind  the 

cornea  there  is  a  thin  mus- 
cular diaphragm,  called  the 
iris,  which  is  visible  in  the 
living  eye  as  its  colored 
part.  The  iris  is  circular  in 
form  and  has  a  circular 
opening,  called  the  pupil,  at 
its  center.  It  regulates  the 
amount  of  light  that  enters 
the  eye  by  involuntary  mus- 
cujar  action,  which  enlarges 

FIG.  313- -Horizontal  Cross-section  of    the    pu    y    when    more    light 
the  Right  Eye.  °  . 

is  needed  and  contracts  it 

when  the  light  is  too  strong.  Just  behind  the  iris  is  the 
crystalline  lens,  a  double-convex,  transparent  solid,  made 
up  of  concentric  layers  which  increase  in  density  and 
refractive  power  toward  the  center.  The  cavity  between 


THE  EYE  417 

the  cornea  and  the  lens  is  filled  with  a  watery  liquid, 
called  the  aqueous  humor.  The  large  cavity  back  of  the 
lens  is  filled  with  a  transparent,  jelly-like  substance,  called 
the  vitreous  humor. 

The  rear  half  of  the  eyeball  is  lined  with  the  retina,  a 
semi-transparent  membrane,  which  contains  a  network 
of  nerve  fibers  branching  from  the  optic  nerve.  A  thin, 
black  membrane,  called  the  choroid  coat,  underlies  the  ret- 
ina and  extends  forward  to  the  iris.  This  membrane  ab- 
sorbs all  light  transmitted  by  the  retina  and  all  diffused 
light  within  the  eye,  making  the  eye  a  dark  chamber. 

Light  on  entering  the  eye  is  refracted  by  the  cornea  and 
aqueous  humor  as  by  a 
convex  lens.  The  crys- 
talline lens  adds  to  this 
effect,  since  it  is  a  more 
refractive  medium  than 

either  the  aqueOUS  Or  the     FlG-  3i4-  -The  Rednaljmage  is  Real  and 

vitreous  humor.  The  re- 
sult is  that  the  light  is  focused  on  the  retina,  forming  real, 
inverted  images  of  external  objects  (Fig.  314).  These 
images  in  some  way  affect  the  retina,  and  the  optic  nerve 
carries  the  impression  to  the  appropriate  brain  center,  pro- 
ducing the  sensation  of  sight.  The  purely  physical  part 
of  the  process  is  the  formation  of  the  image  upon  the  retina. 
If  the  focusing  is  exact  and  the  optic  nerve  is  in  normal  con- 
dition, vision  is  perfect;  if  for  any  reason  the  image  is  more 
or  less  out  of  focus,  vision  is  correspondingly  imperfect. 

The  question  naturally  arises  how  we  see  objects  erect  when  the 
images  in  the  eye  are  inverted.  This  is  only  a  part  of  the  larger 
question  how  an  image  within  the  eye  produces  the  impression  of  an 
external  object,  whether  erect  or  inverted,  or  of  the  still  larger  question 
how  the  image  causes  vision  at  all.  The  physiologist  examines  and 


4i8  LIGHT 

describes  tne  minute  structure  of  the  retina  and  traces  the  course 
of  the  optic  nerve  to  the  brain;  but  the  question  remains  unanswered. 
We  can  only  say  that  experience  teaches  us  to  locate  each  point  of 
an  object  on  the  axis  of  the  cone  of  light  which  enters  the  eye 
from  it  (see  figure).  Seeing  objects  erect  is  thus  a  necessary 
consequence  of  the  fact  that  they  appear  to  be  out  in  space  and  not 
inside  the  eye. 

337.  The  Field  of  Distinct  Vision.  —  The  entire  region  that  is 
visible  when  the  eyes  are  held  in  a  fixed  position  is  called  the  field  of 
vision.     This  field  extends  at  a  very  wide  angle  from  the  eye;  but 
throughout  nearly  the  whole  of  it  objects  are  seen  very  indistinctly. 
This  can  be  readily  tested  by  looking  steadily  at  one  word  of  a  printed 
page,  while  trying  to  read  the  words  round  about  it.     If  the  words 
are  short,  perhaps  three  or  four  can  be  made  out  with  certainty,  but 
not  more.     The  field  of  distinct  vision  is  surprisingly  small.     We  are 
seldom  conscious  of  the  fact,  however,  for  we  are  accustomed  to  fix 
the  attention  wholly  on  the  spot  at  which  we  are  directly  looking. 
In  looking  attentively  at  a  large  object,  a  rapid  shifting  of  the  eyes 
brings  successive  portions  of  it  into  distinct  view. 

When  we  look  directly  at  a  small  object,  its  image  falls  upon  a 
small  central  area  of  the  retina,  which  is  more  sensitive  to  light  than 
the  rest  of  it.  This  part  of  the  retina  is  known  as  the  yellow  spot, 

on  account  of  its  yellowish  color. 

i 

338.  Adaptation  of  the  Eye  to  Different  Distances.  —  We 

know  that,  as  a  distant  object  approaches  a  convex  lens,  its 
real  image  recedes  from  the  lens,  at  first  slowly,  then  more 
and  more  rapidly  as  the  distance  of  the  object  becomes 
relatively  small.  The  perfect  eye,  when  at  rest,  forms 
distinct  images  of  distant  objects  upon  the  retina.  If  the 
eye  were  not  capable  of  some  form  of  adjustment,  the  focus- 
ing would  remain  sensibly  perfect  for  shorter  distances 
down  to  20  ft.;  but  for  distances  less  than  this  the 
light  would  be  focused  behind  the  retina,  and  objects 
would  appear  less  and  less  distinct  when  brought  'hearer 
the  eye. 


THE  EYE  419 

Since  we  are  able  to  see  both  near  and  distant  objects 
distinctly,  it  is  evident  that  the  eye  is  capable  of  adjust- 
ment for  distance.  This  adjustment  is  known  as  accom- 
modation. Observations  upon  the  eye,  such  as  oculists 


CILIARY  MUSCLE 


FAR  NEAR  CILIARY  PROCESS 

FIG.  315.  —  Accommodation. 

are  able  to  make,  have  shown  that  accommodation  is  ef- 
fected by  the  crystalline  lens,  the  front  surface  of  which 
moves  forward  and  becomes  more  convex  when  near  objects 
are  viewed  (Fig.  315).  This  diminishes  the  focal  length 
of  the  lens,  and  increases  the  distance  of  its  center  from 
the  retina,  both  of  which  changes  assist  in  bringing  the 
image  forward  to  the  retina.  The  adjustment  of  the  lens 
for  near  vision  is  controlled  by  the  involuntary  action  of 
the  ciliary  muscle,  which  surrounds  the  lens  in  the  form  of 
a  ring  (shown  in  cross-section  in  the  figure). 

The  power  of  accommodation  is  limited.  When  an  ob- 
ject is  at  less  than  a  certain  distance,  the  effort  to  focus 
the  eye  upon  it  becomes  tiresome;  and  at  still  shorter  dis- 
tances focusing  is  impossible.  These  distances  vary  con- 
siderably for  different  eyes.  For  perfect  eyes  the  least 
distance  that  is  at  all  comfortable  or  suitable  is  about 
25  cm.  or  10  in.  This  is  commonly  taken  as  the  dis- 
tance of  most  distinct  vision  in  computing  the  magnifying 
power  of  microscopes. 

339.    Angular  Size  of  an  Object.  —  As  the  distance  of 


420  LIGHT 

an  object  decreases,  its  image  on  the  retina  grows  larger 
(Fig.  316),  and  smaller  details  of  the  object  are  reproduced 
A  A,  in  it  just  as  a  large  photo- 

graph shows  more  detail 
than  a  small  one.     It  is 
_  E,  owing  to  the  greater  size 

FIG.  316. -Angular  Size  of  an  Object.        of  the  retinal  image,  and 

not  to  more  exact  focus- 
ing, that  we  see  an  object  more  distinctly  as  its  distance 
decreases. 

The  size  of  the  retinal  image  of  an  object  is  proportional 
to  the  angle  within  which  the  object  is  seen,  as  angle  AOB 
or  A' OB'  (Fig.  316).  This  angle  is  called  the  angular  size 
of  the  object.  For  small  angles,  the  angular  size  of  an  ob- 
ject varies  directly  as  its  actual  size  and  inversely  as  its 
distance  from  the  observer.  The  angular  size  of  the  sun, 
as  seen  from  the  earth,  is  approximately  half  a  degree. 
The  angular  size  of  a  copper  cent  at  a  distance  of  7  ft.  is 
the  same.  Hence,  at  these  respective  distances,  the  ret- 
inal images  of  the  sun  and  the  cent  are  of  equal  size. 

When  an  object  is  brought  within  a  few  inches  of  the 
eye,  the  advantage  of  the  larger  retinal  image  is  more  than 
offset  by  the  disadvantage  of  imperfect  focusing.  An  object 
is  seen  most  clearly  when  its  retinal  image  is  as  large  as  it 
can  be,  while  still  in  perfect  focus.  It  is  then  at  the  least  dis- 
tance of  distinct  vision,  which  we  have  assumed  to  be  25  cm. 

340.  Optical  Defects  of  the  Eye.  —  In  some  eyes  the  image  of 
distant  objects  is  formed  in  front  of  the  retina,  the  eyeball  being  too 
long  or  the  curvature  of  the  cornea' or  the  lens  too  great.  Such  eyes 
are  said  to  be  near-sighted,  for  the  image  is  in  focus  upon  the  retina 
only  when  the  object  is  very  near.  This  defect  is  corrected  by  wear- 
ing concave  glasses,  which  offset  the  excessive  convergence  within 
the  eyes  by  increasing  the  divergence  of  the  incident  light. 


THE  EYE 


421 


In  some  cases  the  eye  is  too  short  or  the  crystalline  lens  not  suffi- 
ciently converging,  and  the  focus,  even  for  distant  objects,  would 
fall  behind  the  retina  if  the  power  of  accommodation  were  not  exer- 
cised. Such  eyes  are  far-sighted,  and  cannot  be  focused  on  near 
objects  without  fatiguing  effort,  if  at  all.  Convex  lenses  correct 
the  defect  by  supplementing  the  deficient  convergence  within  the 
eyes. 

In  old  age  the  crystalline  lens  loses  its  elasticity  and  becomes  in- 
capable of  accommodation  for  near  vision.  Hence  old  people,  whose 
eyes  were  perfect  in  earlier  years, 
see  distant  objects  distinctly,  but 
require  convex  glasses  for  reading. 

An  unequal  curvature  of  the 
cornea  or  of  the  crystalline  lens 
in  different  planes  is  called  astig- 
matism. Owing  to  this  defective 
curvature,  the  light  from  any 
point  of  an  object  does  not  con- 
verge to  a  point  on  the  retina, 
but  forms  a  line  instead.  An 
astigmatic  eye  can  not  be  exactly 
focused  for  vertical  and  horizon- 
tal lines  at  the  same  time;  hence 
Fig.  317  presents  a  simple  test 
for  this  defect.  The  radiating 
lines  are  all  alike;  but  to  most 
persons  they  will  appear  unequally  distinct,  for  there  are  very  few 
eyes  that  are  not  astigmatic  in  some  degree.  If  the  difference  in 
distinctness  is  very  marked,  the  eyes  are  strongly  astigmatic,  and 
glasses  should  be  worn,  especially  for  reading  and  other  close  work. 
Astigmatism  is  corrected  either  by  sphero-cylindrical  or  toric  lenses. 
A  sphero-cylindrical  lens  has  a  spherical  curve  on  one  side  and 
a  cylindrical  curve  on  the  other.  The  curvature  of  a  cylindrical 
surface  varies  from  zero  in  the  direction  of  the  axis  of  the  cylinder 
to  a  maximum  at  right  angles  to  the  axis,  and  thus  offsets  the  un- 
equal curvature  of  an  astigmatic  eye.  One  disadvantage  of  this 
lens  is  that  it  is  practically  flat,  like  c  and  e  of  Fig.  301,  whereas 
a  deeply  curved  or  periscopic  lens,  like  b  and  /  of  the  figure,  gives 
a  much  better  field  of  view,  owing  to  the  fact  that  its  entire  surface 


FIG.  317.  —  Test  for  Astigmatism. 


422  LIGHT 

is  nearly  perpendicular  to  the  line  of  sight.  The  toric  lens  is  a 
periscopic  lens,  like  b  and/ of  Fig.  301,  ground  to  correct  astigmatism. 
The  curvature  of  one  surface  is  spherical,  while  that  of  the  other 
varies,  the  minimum  and  maximum  curvatures  being  in  planes  at 
right  angles  to  each  other.  The  grinding  of  such  surfaces  requires 
special  machines,  which  have  only  recently  been  perfected  after 
many  years  of  effort. 

341.  Care  of  the  Eyes.  —  Defective  eyesight  is  very  common,  and 
yet  to  a  large  extent  avoidable.  When  any  defect  is  known  to  exist 
or  is  suspected,  an  oculist  should  be  consulted.  If  it  is  found  that 
glasses  are  needed,  they  should  be  worn;  for  an  uncorrected  defect 
tends  to  become  aggravated,  especially  when  the  eyes  are  much  used 
for  near  work,  as  in  reading.  Even  with  perfect  eyes,  it  is  necessary 
to  exercise  intelligent  care  in  order  to  keep  them  so. 

In  reading,  the  distance  of  the  printed  page  should  not  be  less  than 
14  in.  If  it  must  be  held  closer  than  this  to  be  seen  distinctly,  the 
eyes  are  near-sighted,  and  they  should  be  fitted  with  glasses  for 
constant  use.  When  the  object  viewed  is  held  too  near,  the  eyes  are 
turned  toward  each  other  at  an  excessive  angle  of  convergence, 
and  the  muscular  tension  necessary  to  hold  them  in  this  position 
gradually  pulls  the  eyeballs  out  of  shape,  making  the  sight  still 
more  defective. 

To  one  who  reads  much,  the  proper  illumination  of  the  printed 
page  is  a  very  important  matter,  ignorance  or  neglect  of  which  is 
often  responsible  for  serious  and  permanent  injury  to  the  eyes.  Direct 
sunlight  is  too  intense  for  reading,  and  should  be  avoided.  Artificial 
sources  of  light  should  be  such  as  to  give  a  constant  and  uniform  illu- 
mination, neither  too  faint  nor  too  bright.  A  flickering  light  is  very 
fatiguing.  The  printed  page  is  seen  only  by  the  light  that  it  diffuses. 
The  bright  glare  from  smooth,  glossy  paper  is  due  to  regular  reflection, 
and  is  a  hindrance  to  clear  seeing,  as  experience  teaches.  Glare  is 
disastrous  to  the  eyes,  and  should  not  be  tolerated  for  a  moment. .  It 
can  be  avoided  by  holding  the  page  at  such  an  angle  that  the  regularly 
reflected  light  is  thrown  off  to  one  side. 

One  should  not  sit  facing  the  source  of  light,  even  if  it  is  covered 
with  a  shade;  and,  if  it  is  bare,  it  should  by  all  means  be  out  of  the 
range  of  vision.  It  is  difficult  to  get  a  good  distribution  of  light, 
either  for  reading  or  writing,  from  a  table  lamp  of  any  description. 


THE  EYE  423 

An  electric  lamp,  pointed  directly  downward,  or  nearly  so,  from  a 
low  chandelier  or  a  wall  bracket  answers  all  requirements.  The 
lamp  should  be  covered  with  a  reflector  or  a  shade,  designed  to  con- 
centrate the  light  in  a  downward  direction. 

Lastly,  the  eyes  should  be  exposed  as  little  as  possible  to  the 
undiffused  light  from  brilliant  sources,  such  as  incandescent  lamps, 
arc  lights,  and  Welsbach  burners.  Strong  light  should  always  be 
diffused  and  softened  by  the  use  of  frosted  bulbs  and  globes,  reflec- 
tors, shades,  etc. 

342.  Binocular  Vision.  —  The  ordinary  use  of  both  eyes  at  the 
same  time  is  called  binocular  vision.  A  person  who  has  two  eyes 
rarely  uses  one  alone,  except  for  some  special  purpose,  as  in  aiming 
a  gun;  and  he  is  therefore  likely  to  be  wholly  unaware  of  the  interest- 
ing and  important  differences  between  vision  with  one  eye  and  with 
both.  The  principal  differences  are  shown  in  the  following  simple 
experiments. 

Hold  a  pencil,  point  up,  at  a  distance  of  12  or  14  in.  from  the 
eye,  and  a  second  pencil,  point  down,  at  arm's  length.  With  one  eye 
closed,  bring  the  points  of  the  two  pencils  into  line  with  the  other  eye. 
They  now  appear  to  touch  each  other,  although  they  are  nearly  a  foot 
apart.  The  single  eye  conveys  no  impression  of  the  unequal  distances. 
Look  with  both  eyes,  and  you  at  once  receive  the  true  impression  of 
distance. 

Holding  the  pencils  as  before,  in  line  with  one  eye,  close  that  eye 
and  look  with  the  other.  The  pencils  no  longer  appear  to  be  in  line. 
With  the  two  eyes  we  see  the  same  object  from  two  slightly  different 
positions,  and  hence  in  slightly  different  directions.  To  study  this 
further,  hold  up  a  finger  before  your  face  and  look  beyond  it,  with 
both  eyes,  at  a  wall.  The  finger  appears  double  and  transparent. 
(Explain.) 

Distances  to  right  and  left  and  distances  up  and  down  are  per- 
ceived with  one  eye  as  well  as  with  two;  but  the  impression  of  distance 
along  the  line  of  sight  is  much  more  vivid  and  accurate  when  both 
eyes  are  used.  This  is  mainly  due  to  the  fact  that  the  eyes  are  turned 
toward  each  other  more  or  less,  according  to  the  distance  of  the 
point  at  which  they  are  directed  (Fig.  318).  By  experience  we  learn 
unconsciously  to  base  our  estimate  of  distance  on  the  greater  or  less 
convergence  of  the  lines  of  sight,  BA  and  CA,  of  the  two  eyes. 


424  LIGHT 

Another  reason  why  the  impression  of  distance  is  more  definite 
in  binocular  vision  is  that,  with  the  two  eyes,  we  have  two  slightly 

different  views  of  an  object  at 
the  same  time.  In  looking  at 
a  small  cube,  for  example,  we 

can  see  the  fr°nt  anc*  tlie  "S^t 
side  with  the  right  eye  and,  at 
the  same  time,  the  front  and  the  left  side  with  the  left  eye.  It 
is  to  these  dissimilar  views  that  we  owe  the  mental  impression 
of  solidity  or  of  form  in  three  dimensions.  Let  us  try  to  analyze 
this  impression.  When  we  look  at  an  object  with  both  eyes,  the  point 
of  it  to  which  the  eyes  are  at  any  instant  directed  appears  single,  while 
all  the  rest  of  it  appears  double.  Ordinarily  we  are  not  conscious  of 
this  doubleness,  for  the  attention  is  fixed  on  the  point  under  direct 
observation;  but  if  we  direct  the  attention  to  the  whole  object  while 
looking  steadily  at  one  point,  the  apparent  doubling  is  very  conspicu- 
ous. Thus  when  we  look  at  a  long  pencil,  held  in  the  hand  with  the 
sharp  end  pointing  toward  the  chin,  it  appears  as  shown  in  a,  b,  or  c 
of  Fig.  319,  according  as  the  eyes  are  directed 
toward  the  nearer  end,  the  middle,  or  the 
farther  end  of  it.  The  impression  of  distinct 
vision  for  the  point  under  direct  observation 
and  the  impression  of  indistinct  and  double 
vision  for  the  remainder  of  the  object  together 
make  up  the  impression  of  solidity  or  of  form 
in  the  three  dimensions  of  space.  With  a  single  eye  there  is  only 
an  imperfect  suggestion  of  the  third  dimension,  or  distance  from 
the  observer,  as  in  a  photograph. 

343.  The  Principle  of  the  Stereoscope.  —  If  we  present  to  each 
eye  a  picture  of  an  object  taken  from  its  point  of  view,  and  direct 
the  eyes  so  that  these  slightly  dissimilar  pictures  seem  to  coincide, 
the  appearance  of  solidity  will  be  perfectly  reproduced.  This  can  be 
shown  with  Fig.  320.  The  two  pictures  of  the  tunnel  represent  it 
as  it  is  seen  by  the  two  eyes  separately.  Hold  the  book  so  that  one 
picture  is  immediately  in  front  of  each  eye,  with  a  card  between 
them  and  perpendicular  to  the  page,  so  that  each  eye  can  see  only 
the  picture  on  its  own  side.  Now  direct  the  eyes  as  if  you  were 
looking  through  the  book  at  a  point  some  distance  behind  it.  When 


OPTICAL    INSTRUMENTS 


425 


this  is  done,  the  pictures  will  seem  to  move  together  and  unite  into 
a  single  view,  which  has  the  appearance  of  real  depth  extending 
2  ft.  or  more  behind  the  book.  This  stereoscopic  picture  will 


FIG.  320.  —  Stereoscopic  Pictures. 

appear  blurred  for  half  a  minute  or  more  while  the  eyes  are  strug- 
gling to  bring  it  into  focus,  but  it  finally  becomes  perfectly  clear. 
With  a  little  practice,  the  card  between  the 
pictures  can  be  dispensed  with;  but  two  ad- 
ditional tunnels  will  then  be  indistinctly  seen, 
one  on  either  side,  the  one  on  the  left  being 
the  left  picture  as  seen  by  the  right  eye  and  the 
other  the  right  picture  as  seen  by  the  left  eye. 
The  stereoscope  is  an  instrument  designed 
to  aid  the  eyes  in  uniting  into  one  view  two 
slightly  dissimilar  photographs  of  the  same 
scene.  These  photographs  are  taken  with 
a  double  camera,  and  represent  the  scene 
just  as  it  would  appear  to  the  two  eyes  of 
an  observer.  They  are  mounted  on  the 
same  card,  AB  (Fig.  321),  and  are  viewed 
through  the  half  lenses  M  and  N.  The 
magnified  virtual  images  of  the  two  pictures 
coincide  at  A'  B' .  The  partition  P  prevents  each  eye  from  seeing 
the  picture  intended  for  the  other. 

VII.  OPTICAL  INSTRUMENTS 

344.   Magnification.  —  In  the  elementary  study  of  opti- 
cal instruments,  such  as  microscopes,  telescopes,  and  opera 


-*' 

/>           p 
i           j> 

1 

| 

i 

\ 

i 
i 

\ 

\ 

j 

\ 

r 

\ 

i 

\ 

'A                '  R 

A\          /?  - 

/ 

\ 
\ 

' 

/ 

I 

/    P 

i 

\ 

i 

l 

1 

1 

i>  ' 

<d 

N 

FIG.    321.  —  Diagram  of 

the  Stereoscope. 

426  LIGHT 

glasses,  we  are  principally  concerned  with  the  two  ques- 
tions (i)  how  the  instrument  forms  the  image  that  is  seen 
in  looking  through  it,  and  (2)  how  great  is  the  advantage 
gained  by  its  use. 

The  one  general  purpose  of  all  such  instruments  is  to 
form  an  enlarged  image,  which  can  be  viewed  to  better 
advantage  than  the  object  itself.  The  proper  measure  of 
this  advantage  is  the  ratio  of  the  size  of  the  retinal  image 
with  the  instrument  to  its  size  when  the  object  is  viewed 
with  the  naked  eye.  This  ratio  (of  linear  dimensions, 
not  of  areas)  is  called  the  magnifying  power  of  the  instru- 
ment. Since  the  size  of  the  retinal  image  is  always  propor- 
tional to  the  angular  size  of  its  real  or  apparent  source  (Art. 
339),  the  magnifying  power  of  an  optical  instrument  is  de- 
nned as  the  ratio  of  the  apparent  angular  size  of  an  object 
when  viewed  through  the  instrument  to  its  angular  size 
when  viewed  with  the  naked  eye. 

345.  The  Simple  Microscope.  —  The  converging  lens 
is  much  used  as  a  simple  microscope  in  viewing  small  ob- 
jects. What  the  observer  sees  is  a  magnified  virtual  im- 


FIG.  322.  —  The  Simple  Microscope. 

age  of  the  object  (Fig.  322).  Lenses  for  this  purpose  are 
mounted  in  frames  and  supports  of  various  forms.  The 
pocket  magnifier  and  the  reading  glass  are  familiar  examples. 
The  angular  size  of  the  image,  A'B'\  is  greatest  when  the 
eye  is  close  to  the  lens  and  the  image  is  at  the  least  dis- 


OPTICAL  INSTRUMENTS  427 

tance  of  distinct  vision,  or  25  cm.  With  this  adjustment, 
the  angular  size  of  the  image  is  the  angle  A'OB',  and  EO 
=  25  cm.  When  the  object  is  viewed  with  the  naked  eye, 
its  distance  can  not  be  less  than  EO,  and  its  angular 
size  is  then  angle  A^OB^.  Hence  the  magnifying  power 

of  the  lens  is  the   ratio  — «p-  -  by   definition.     For 

angle  A2OB2 

A'B'       A'Bf 
small  objects  this"  is  equal  to  -T-^  or  -j-^r  and,  from  the 

AZ&Z  A.\&\ 

A  /  Ttf  Tff) 

similar  triangles  A'OB'  and  AiOBi,  -7—=r  =  -=-=•• 

A\D\        L/U 

Simple  magnifiers  are  always  lenses  of  short  focal  length 
(generally  from  2  to  6  cm.) ;  and  for  such  lenses  the  focal 
length,  /,  is  only  slightly  greater  than  the  distance  of  the 
object,  DO,  and  may  be  substituted  for  it  in  the  above 
ratio.  Hence  the  magnifying  power  of  a  simple  micro- 

2>t 

scope  is  approximately  «rj?  when  the  unit  of  length  is  the 

v  •* 

centimeter,  and  -j  when  the  unit  is  the  inch. 

The  usual  range  of  magnifying  power  for  single  lenses  is  from  5 
to  10  diameters.  With  higher  powers  the  image  is  badly  distorted 
and  colored,  owing  to  spherical  and  chromatic  aberration.  Doublet 
magnifiers  give  good  results  with  powers  as  high  as  24  diameters. 
These  have  two  lenses  placed  a  short  distance  apart. 

346.  The  Compound  Microscope.  —  In  viewing  very 
minute  objects  a  higher  power  is  required  than  is  possible 
with  a  simple  magnifier.  The  instrument  designed  for 
this  purpose  is  called  a  compound  microscope.  In  its  ele- 
mentary form  it  consists  of  two  convex  lenses  of  short  focal 
length,  called  the  objective,  O,  and  the  eye-lens,  E  (Fig. 
323).  The  objective  is  placed  at  a  distance  only  slightly 
greater  than  its  focal  length  from  the  object,  AB}  in  order 


428  LIGHT 

to  form  a  magnified  real  image,  A'B',  at  a  much  greater 
distance  on  the  other  side.  The  eye-lens  serves  the  pur- 
pose of  a  simple  microscope,  through  which  the  observer 
sees  a  magnified  virtual  image,  A"B" ,  of  the  real  image. 

Let  DI  denote  the  distance  of  the  real  image  from  the 
objective.     The  distance  of  the  object  may  be  taken  as 


FIG.  323.  —  Diagram  of  the  Compound  Microscope. 

the  focal  length,  /0,  of  the  objective.  The  magnification 
due  to  the  objective  is  the  ratio  of  the  size  of  the  real  image 
to  the  size  of  the  object,  and  this  is  equal  to  the  ratio 

--1.     The  magnifying  power  of  the  eye-lens  is  — ,  /e  being 

Jo  .  7e     ' 

its  focal  length.  The  magnifying  power  of  the  two  lenses 
together  is  the  product  of  their  separate  magnifying 

powers,  or    f  f  ',  all  distances  being  expressed  in  inches, 
yeyo 

The  distance  D\  is  determined  by  the  length  of  the  microscope 
tube,  at  the  ends  of  which  the  lenses  are  inserted  (Fig.  324).  In 
standard  instruments  it  is  approximately  6  in.  As  shown  in  the 
formula,  the  magnification  varies  inversely  as  the  focal  length  of 
either  the  objective  or  the  eye-lens.  Hence  the  shorter  the  focal 
length  of  either,  the  greater  is  the  magnification.  In  ordinary 


OPTICAL  INSTRUMENTS 


429 


practice  the  objectives  vary  in  focal  length  from  f  in.  to  i  in.;  and 
with  these,  both  two-inch  and  one-inch  eyepieces  are  used.  With 
a  £  objective  and  a  one-inch  eyepiece,  the  magnification  is 

—  =  360  diameters.     Higher  powers,  up   to   2000  diameters, 
1  X  i 

are  possible. 

The  images  formed  by  a  compound  microscope  consisting  of 
two  single  lenses  are  distorted  and  indistinct,  and  are  colored  by 
chromatic  aberration.  To  avoid  these  defects  the  objective  is  built 
up  of  from  four  to  ten  lenses  (Fig.  339),  and  the  eyepiece  of  two. 

347.  The  Astronomical  Telescope.  —  The  telescope 
serves  the  same  purpose  in  viewing  a  distant  object  that 
the  microscope  does  in  viewing 
a  small  one.  Both  instruments 
form  images  which  have  a  larger 
angular  size  than  the  objects. 
There  are  various  types  of  tele- 
scopes, including  such  extremes 
of  size  and  use  as  the  astro- 
nomical telescope  and  the  opera 
glass. 

The  astronomical  telescope,  in 
its  simplest  form,  consists  of  two 
convex  lenses,  which  serve  as 
objective  and  eye-lens  respect- 
ively, as  in  the  compound  micro- 
scope. Since  the  object  viewed 
is  always  a  distant  one  its  real 
image,  ab  (Fig.  325),  is  at  the  FlG- 
principal  focal  distance  of  the 
objective.  The  eye-lens  forms  a  magnified  virtual  image, 
a'b',  of  the  real  image.  Both  the  real  and  the  virtual 
images  are  inverted  with  respect  to  the  object;  but  this 
is  not  a  disadvantage  in  viewing  the  heavenly  bodies. 


Mi' 


430 


LIGHT 


The  object  is  not  shown  in  the  figure,  since,  in  its  true 
relative  position,  its  distance  would  be  enormously  greater 
than  the  focal  length  OC.  The  parallel  lines  JK  and  NO 
represent  rays  from  a  point  at  the  lower  side  of  the  object. 
All  such  rays  converge  to  the  corresponding  point  b  of  the 


FIG.  325.  —  The  Astronomical  Telescope. 

image.  Similarly  the  parallel  rays,  such  as  MO  and  HI, 
from  a  point  at  the  top  of  the  object  converge  to  a.  The 
angular  size  of  the  object  is  the  angle  MON,  or  the  equal 
angle  aOb.  The  angular  size  of  the  image,  as  it  appears 
in  looking  through  the  telescope,  is  the  angle  a'Ebf,  or 
the  equal  angle  aEb.  Hence  the  magnification  is,  by 

angle   aEb 

definition,  -          — —  •     Since  these  angles  are  subtended 
angle   aOb 

by  the  same  line  ab,  they  are  (for  small  angles)  inversely 
proportional  to  the  distances  of  their  vertices  from  this 

line;  i.e.  -  = Now  OC  is  the  focal  length, 

angle  aOb     CE 

/0,  of  the  objective,  and  CE  is  sensibly  equal  to  the  focal 
length,  /j,  of  the  eye-lens.  Hence  the  magnifying  power 

of  the  telescope  is  A  or  the  ratio  of  the  focal  length  of 

7i 

the  objective  to  the  focal  length  of  the  eye-lens.  It 
should  be  noted  that  this  ratio  is  increased  either  by  in- 
creasing /0  or  by  decreasing  f{.  The  objectives  of  the  most 


OPTICAL    INSTRUMENTS  431 

powerful  telescopes  have  focal  lengths  ranging  from  40  to 
60  ft.  or  more.  The  eyepieces  are  the  same  as  in  com- 
pound microscopes,  and  serve  exactly  the  same  purpose. 

An  objective  of  great  focal  length  must  also  have  a  large  diameter; 
for  a  highly  magnified  image  will  not  be  as  bright  as  is  necessary 
unless  it  is  formed  by  a  proportionately  great  amount  of  light,  and  the 
light-gathering  power  of  the  objective  is  proportional  to  its  area. 
The  great  telescope  of  the  Lick  Observatory,  at  Mt.  Hamilton, 
California,  has  a  diameter  of  36  in.  and  a  focal  length  of  57  ft.; 
that  of  the  Yerkes  Observatory,  at  Williams  Bay,  Wisconsin,  a  diam- 
eter of  40  in.  and  a  focal  length  of  62  ft.  The  largest  telescope 
objective  ever  constructed  is  that  of  the  Carnegie  Solar  Observatory, 
on  the^ummit  of  Mt.  Wilson,  near  Pasadena,  California.  This  lens 
is  60  in.  in  diameter  and  is  8  in.  thick  at  the  center.  Three 
years  were  spent  in  grinding  and  polishing  its  surfaces.  The  great 
telescope  of  the  Paris  Exposition,  in  1900,  was  180  ft.  long,  and  its 
objective  47  in.  in  diameter.  Owing  to  its  enormous  size,  it  was 
mounted  in  a  fixed  horizontal  position,  and  the  light  from  the  heavenly 
bodies  was  reflected  into  it  by  a  large  mirror. 

With  such  telescopes  as  these  the  surface  of  the  moon  is  shown  in 
great  detail,  objects  less  than  half  a  mile  in  diameter  being  visible, 
and  the  earth's  nearer  neighbors  —  Mars,  Jupiter,  and  Saturn  — 
appear  as  large  and  beautiful  orbs.  The  fixed  stars,  however,  are 
at  such  enormous  distances  that  they  are  seen  only  as  points  of  light; 
but  the  telescope  increases  their  brightness  and  the  apparent  distance 
between  them,  and  thus  brings  into  view  millions  of  stars  which  are 
never  seen  with  the  naked  eye.  Even  an  opera  glass  greatly  increases 
the  number  of  visible  stars. 

348.  The  Terrestrial  Telescope  has  two  additional  lenses  to  rein- 
vert  the  image  and  make  it  erect.  A  single  lens,  L  (Fig.  326),  would 
accomplish  this  result,  but  the  image  would  be  imperfect.  The 
objective  (not  shown  in  the  figure)  forms  the  inverted  real  image  ab. 
The  lens  L,  placed  at  twice  its  focal  length  from  this  image,  forms  a 
real  and  erect  image,  a'b',  at  an  equal  distance  on  its  opposite  side; 
and  the  eyepiece  forms  an  erect  virtual  image  of  a' b'.  A  small 
terrestrial  telescope  is  called  a  spy-glass. 


432 


LIGHT 


349.  Opera  and  Field  Glasses.  —  In  the  common  opera 
glass  or  field  glass  the  part  for  each  eye  is  a  complete  tele- 
scope of  the  type  shown  in  Fig.  327.  The  objective  is  a 
convex  lens,  having  a  focal  length,  OC,  of  about  4  in.  in 


FIG.  326.  —  The  Terrestrial  Telescope. 

opera  glasses  and  5  to  7  in.  in  field  glasses.  The  eye-lens 
is  concave  and  of  very  short  focal  length,  EC.  The  dis- 
tance between  the  lenses  is  equal  to  or  very  slightly  less 
than  the  differences  between  their  focal  lengths. 

The  objective  would  form  an  inverted  real  image,  ab,  of 
a  distant  object  if  the  light  were  not  intercepted  by  the  eye- 


FIG.  327. —  The  Galilean  Telescope. 

lens;  but,  with  this  lens  in  position,  the  converging  cone 
of  light  from  any  point  of  the  object  is  changed  to  a  slightly 
diverging  cone,  and  forms  a  virtual  image  of  the  point. 
Thus  all  rays  converging  toward  a  focus  at  a  appear  to 


OPTICAL  INSTRUMENTS  433 

come  from  a',  and  all  rays  converging  toward  b  appear 
to  come  from  b'.  Hence  the  only  image  actually  formed 
is  the  erect  virtual  image  a'br . 

The  angular  size  of  this  image  is  the  angle  a'Eb'  or  its 
equal  a£6,.and  the  angular  size  of  the  object  is  MON  or 
its  equal  aOb.  The  magnifying  power  of  the  instrument  is 

therefore  angle  aEl ',  which  is  equal  to**?,  or  the  ratio  of 
angle  aOb  EC 

the  focal  length  of  the  objective  to  the  focal  length 
of  the  eye-lens,  as  for  the  astronomical  telescope.  For 
equal  power  the  opera  glass  is  the  shorter  instrument 
by  twice  the  focal  length  of  the  eye-lens.  It  has  a  still 
greater  advantage  in  this  respect  over  the  terrestrial  tele- 
scope, since  the  lenses  in  the  latter  for  erecting  the  image 
increase  the  length  without  increasing  the  power. 

The  earliest  telescopes  were  constructed  on  the  principle  of  the 
opera  glass;  but  they  were  single-tube  instruments,  for  one  eye  only. 
The  first  authentic  record  of  such  an  instrument  is  of  one  made  in 
Holland  in  1608.  Galileo,  hearing  of  this  invention,  took  up  the  prob- 
lem, and  was  soon  making  telescopes  of  considerable  power.  With 
their  aid  he  made  several  astronomical  discoveries  which  occasioned 
a  great  sensation  at  the  time  and  brought  him  great  renown.  "  He 
turned  his  telescope  toward  the  moon  and  discovered  mountains  and 
craters;  he  turned  it  to  Jupiter  and  saw  its  satellites;  he  pointed  it 
at  Saturn  and  saw  the  planet  threefold  (now  known  to  have  been  due 
to  an  imperfect  view  of  the  ring) ;  he  examined  the  sun,  saw  its  spots 
moving,  and  concluded  that  the  sun  rotates."  The  early  history  of 
the  telescope  is  therefore  closely  associated  with  the  name  of  Galileo; 
and  the  telescope  with  a  concave  eye-lens  is  commonly  known  as 
Galileo's  telescope. 

350.  The  Prism  Binocular.  —  There  is  one  important  matter 
concerning  telescopes  of  all  kinds  which  remains  to  be  considered, 
namely,  the  size  of  the  field  of  view.  The  field  of  an  opera  glass  of 
the  common,  or  Galilean,  type  is  large  enough  to  include  only  a  small 


434 


LIGHT 


group  of  actors,  and  it  is  necessary  to  turn  the  glass  about  in  order 
to  see  the  different  parts  of  the  stage.  With  different  telescopes 
of  the  same  type  the  angular  diameter  of  the  field  of  view  varies 

inversely  as  the  magnifying 
power.  In  doubling  the 
power  the  diameter  of  the 
field  is  reduced  one  half,  and 
its  area  is  reduced  fourfold. 
In  considering  the  relative 
merits  of  the  different  types 
of  instruments,  the  magnify- 
ing power  and  the  size  of  the 
field  must  both  be  taken  into 
account.  A  terrestrial  tele- 
scope has  a  much  larger  field 
than  a  Galilean  telescope  of  the  same  power,  but  it  is  much  less  con- 
venient on  account  of  its  greater  length  and  weight.  A  new  type  of 
instrument  has  recently  come  into  use,  which  combines  the  advantages 
of  a  large  field  and  a  short,  compact  form.  This  is  the  prism  bin- 


FIG.  328.  —  The  Stereo-Prism  Binocular. 


FIG.  329.  —  Magnifying  Power. 

ocular  (Fig.  328).  The  large  field  is  secured  by  its  converging  eye- 
piece, which  is  the  same  as  in  astronomical  and  terrestrial  telescopes. 
Its  small  size  is  due  to  the  use  of  two  total-reflecting  prisms  in  each 
tube.  A  ray  of  light  is  reflected  twice  in  the  same  plane  at  one  prism, 


OPTICAL  INSTRUMENTS 


435 


and  twice  in  a  plane  at  right  angles  to  the  first  at  the  second  prism. 
The  angle  of  incidence  at  each  reflection  is  45°;  hence  the  image  is 
turned  through  an  angle  of  90°  at  each  reflection.  These  four  reflec- 
tions, therefore,  reinvert  and  reverse  the  image  (interchanging  top  and 
bottom,  and  right  and  left  sides),  thus  accomplishing  the  same  result 
as  the  erecting  lens,  L  (Fig.  326),  of  the  terrestrial  telescope.  But 
while  the  erecting  lens  increases  the  length  of  the  instrument,  the  prism 
shortens  it  nearly  two  thirds;  for  the  tube,  although  comparatively 
long,  is  folded  into  three  parts  lying  side  by  side.  A  prism  binocular 
having  a  field  of  the  same  size  as  that  of  an  old-style  glass  has  three 
times  the  magnifying  power  (Fig.  329). 

In  the  stereo-prism  binocular,  shown  in  Fig.  328,  the  objectives 
are  considerably  farther  apart  than  the  eyes.  This  wider  separation 
of  the  two  points  of  view  enhances  the  stereoscopic  effect  of  binocular 
vision. 

351.  The  Projection  Lantern.  —  With  this  instrument  highly 
magnified  images  of  transparent  photographs  and  drawings  are  pro- 
jected on  a  white  screen  in  a  dark  room.  Its  essential  parts  are  a 
strong  source  of  light,  A  (Fig.  330),  a  condensing  lens  or  pair  of  lenses, 


FIG.  330.  —  The  Projection  Lantern. 

L,  for  concentrating  the  light  upon  the  picture  or  "  slide  "  P,  and  an 
achromatic  objective,  Lf,  placed  at  a  little  more  than  its  focal  length 
from  the  slide.  The  image  is  formed  at  a  relatively  great  distance, 
and  is  correspondingly  magnified.  In  order  that  it  may  be  erect, 
the  slide  is  inverted. 

352.   The  Biograph,  or  moving-picture  machine,  is  a  projection 
lantern  with  a  mechanism  by  which  a  series  of  pictures,  printed  on  a 


436  LIGHT 

long  film,  can  be  thrown  upon  a  screen  in  rapid  succession.  The 
pictures  follow  one  another  at  the  rate  of  about  15  or  20  per  second; 
and,  -as  each  comes  into  position  in  the  instrument,  a  flash  of  light 
projects  it  upon  the  screen.  During  the  intervals  between,  while  the 
film  is  shifting  from  one  picture  to  the  next,  the  light  is  cut  off  by  a 
diaphragm,  and  the  screen  is  then  dark.  The  picture  appears  to 
be  continuously  present  on  the  screen,  for  the  visual  impression 
persists  long  enough  to  bridge  over  the  interval  of  darkness.  (It 
is  owing  to  this  "  persistence  of  vision  "  that  falling  rain-drops  are 
seen  as  long  streaks,  vibrating  strings  as  gauzy  spindles,  etc.) 

Moving  picures  are  taken  with  cameras  provided  with  a  mecha- 
nism which  opens  and  closes  the  shutter  at  regular  intervals,  and 
which,  while  the  shutter  is  closed,  jerks  the  ribbon  film  into  place  for 
the  next  impression.  As  the  pictures  taken  are  of  moving  objects, 
they  record  successive  positions  of  the  objects  at  equal  time  inter- 
vals; and,  when  they  are  projected  upon  a  screen,  the  objects 
appear  to  move  from  each  position  to  the  next,  owing  to  the  persist- 
ence of  the  visual  impression. 


PROBLEMS 

1.  What  is  the  magnifying  power  of  a  simple  microscope  whose  focal 
length  is  (a)  2  in.?    (b)  .5  in.? 

2.  Show  from  the  formula  of  Art.  333  that  the  magnifying  power  of 
a  simple  microscope  is  inversely  proportional  to  its  focal  length.     Draw 
figures  to  illustrate. 

3.  Assuming  that  the  real  image  is  formed  5  in.  from  the  objective,  find 
the' magnifying  power  of  a  compound  microscope  (a)  with  a  -5-in.  objective 
and  a  2-in.  eyepiece;   (b)  with  a  |-in.  objective  and  a  £-in.  eyepiece. 

4.  The  great  telescope  of  the  Lick  Observatory  is  57  ft.  long.     What 
is  its  magnifying  power  (a)  when  fitted  with  a  2-in.  eyepiece?    (b)  when 
fitted  with  a  ^-in.  eyepiece? 

6.  The  objective  of  a  field  glass  has  a  focal  length  of  7.5  in.,  and  the 
eyepiece  a  focal  length  of  1.25  in.  Find  the  length  of  the  instrument  and 
its  magnifying  power. 

6.  What  are  the  essential  parts  of  a  photographic  camera,  and  what 
purpose  do  they  serve?  What  adjustment  is  necessary  for  the  distance  of 
the  object  and  why?  How  is  the  size  of  the  image  of  any  object  affected 


OPTICAL  INSTRUMENTS  437 

by  the  distance  of  the  object  from  the  camera?  Why  does  a  large  camera 
take  a  larger  picture  of  a  distant  object  than  a  small  camera  does?  How 
is  the  necessary  time  of  exposure  affected  by  "stopping  down"  the  lens 
opening  with  the  diaphragm? 

Suggestion.  —  If  you  have  a  camera  you  will  be  interested  in  trying 
"pinhole  photography."  For  this  purpose  remove  the  lens  and  cover  the 
opening  with  tinfoil.  In  the  center  of  the  foil  make  a  minute  hole  with  a 
fine  needle.  The  edge  of  the  hole  should  be  as  smooth  and  thin  as  possible. 
How  would  you  compute  the  proper  time  of  exposure,  knowing  the  time 
necessary  with  the  lens  and  a  given  adjustment  of  the  diaphragm? 


VIII.    DISPERSION  OF  LIGHT.    COLOR 

353.  The  Composite  Character  of  Sunlight.  Disper- 
sion. —  When  a  beam  of  sunlight  is  admitted  through  a 
small  opening  into  a  dark  room  and  allowed  to  fall  on  a 
white  screen,  at  a  distance  of  several  meters  from  the  open- 
ing, it  forms  on  the  screen  a  brilliant  pinhole  image  of  the 
sun.  A  prism  placed  in  the  path  of  the  beam  deflects 
it  to  one  side.  When  this  deflected  beam  falls  on  the 
screen,  it  appears  as  a  band  of  light,  VR  (Fig.  331),  rounded 
at  the  ends  and  brilliantly  colored.  Violet,  blue,  green, 
yellow,  orange,  and  red  are  all  present,  in  the  order 
named.  These  beams  of  colored  light  can  be  brought 
together  again  on  the  screen  by  means  of  a  lens,  placed  at 
conjugate  focal  distances  from  the  prism  and  the  screen 
(Fig.  332),  or  by  means  of  a  second  prism,  placed  near  the 
first  but  with  its  refracting  angle  in  the  opposite  direction 
(Fig.  333) ;  and  the  round  spot  on  which  the  light  falls  is 
white. 

What  -is  the  lesson  conveyed  by  these  strikingly  beauti- 
ful experiments?  Obviously  a  prism  or  a  lens  of  colorless 
glass  can  not  make  or  destroy  colored  lights.  The  plain 
inference  is  that  these  colored  lights  are  present  in  the  sun- 
beam and  are  separated  by  unequal  refraction  in  passing 


438 


LIGHT 


through  the  first  prism,  and  that  the  second  prism  or  the 
lens  brings  them  together  again  in  the  same  condition  as  at 
first.  The  white  light  of  the  sun  consists  of  these  various 
colored  lights,  all  traveling  together.  Their  separation  by 


FIG.  331.  —  Dispersion  by  Prism;  Impure  Spectrum. 

the  prism  is  called  dispersion,  and  the  prism  is  said  to 
analyze  the  sunbeam  into  its  constituent  colors,  forming 
the  solar  spectrum.  Similarly,  when  the  light  from  any 
source  is  separated  into  its  constituents  by  a  prism,  or  by 


FIG.  332.  —  Colors  of  Spectrum  Recombined  by  Lens. 

other  means,  the  resulting  colors,  taken  together,  are  called 
the  spectrum  of  that  light. 

The  solar  spectrum,  when  formed  as  described  above, 
consists  of  an  indefinitely  great  number  of  colored  images 
of  the  sun,  overlapping  one  another.  If  a  piece  of  red 
(ruby)  glass  is  placed  in  front  of  the  opening,  it  trans- 
mits only  the  red  light  and  absorbs  the  other  colors.  The 


DISPERSION  OF  LIGHT.     COLOR  439 

spectrum  will  then  consist  of  a  circular  red  spot,  which 
is  the  red  image  of  the  sun.     There  are  thousands  of  such 
images,  of  as  many  different  gradations  of  color,  in  the 
complete  spectrum,  and  there  is  consequently  much  over- 
lapping and  mixing  of  the  colors.     A  spectrum  of  this  char- 
acter is  said  to  be  imperfect  or  impure. 

On  looking  through  the  prism,  with  the  eye  close  to  it 
and  in  such  a  position  as  to  receive  all  the  colors,  the  ob- 
server will  see  a  virtual  /\r^^ 
spectrum,    V'R'    (Fig.     s    - — —       ^^^^===    ====)» 

331),  which   is  nearly 

pure.        This     is     quite     ^IG-    333>  —  Colors  of    Spectrum   Recombined 

by  Second  Prism. 

satisfactory    for    indi- 
vidual observation  in  the  laboratory  (see  Lab.  Ex.  56); 
but  for  experimental  work  in  the  class-room  the  spectrum 
must  be  real.     It  should  also  be  large,  brilliant,  and  ap- 
proximately pure. 

354.  Formation  of  a  Pure  Spectrum.  Fraunhofer's 
Lines. — To  obtain  a  wide  spectrum  the  sunlight  is  admitted 
through  an  opening  2  or  3  cm.  long.  To  prevent  overlap- 
ping of  the  colored  images  the  opening  must  be  narrow. 
A  long,  narrow  slit  fulfils  both  these  requirements.  It 
should  be  vertical  and  the  sunbeam  horizontal.  To 
obtain  a  pure  spectrum  the  illuminated  slit  must  be  treated 
as  the  source  of  light,  not  the  sun;  and  the  colored  images 
of  the  slit  must  be  exactly  focused  on  the  screen  by  means 
of  a  lens.  To  secure  this  adjustment  the  screen  XY 
(Fig.  334),  is  placed  at  a  distance  of  several  meters  from  the 
slit,  and  the  lens  is  adjusted  so  as  to  form  a  magnified  im- 
age of  the  slit  upon  it,  the  prism  being  removed.  The  prism 
is  next  placed  in  line,  near  the  lens  (on  either  side  of  it), 
and  the  screen  moved  into  the  position  X'Yf,  in  line  with 


440  LIGHT 

the  deflected  light  and  at  the  same  distance  from  the 
lens  as  before.  The  spectrum  thus  formed  consists  of  a 
series  of  narrow  and  only  slightly  overlapping  images  of 
the  slit. 

A  prism  of  flint  glass  produces  a  wider  separation  of  the 
colors  and  a  longer  spectrum  than  one  of  crown  glass,  and 
a  bottle  prism  containing  carbon  bisulphide  is  still  better. 
With  a  suitable  adjustment  of  the  apparatus,  the  carbon 
bisulphide  prism  gives  a  spectrum  a  foot  wide  and  two  feet 
or  more  in  length;  and  in  a  perfectly  dark  room  the 


FIG.  334.  —  Pure  Spectrum  by  Means  of  Lens  and  Prism. 

spectrum  is  very  brilliant.  Under  such  conditions  the 
experiment  is  the  most  beautiful  in  the  whole  range  of 
elementary  physics. 

With  a  bisulphide  prism,  a  very  narrow  slit,  and  a  lens 
exactly  in  focus,  the  spectrum  is  very  pure  and  is  crossed 
by  many  dark  lines  at  right  angles  to  its  length  (i.e.  in  a 
direction  parallel  to  the  slit) .  These  lines  are  called  Fraun- 
hofer's  lines,  after  the  celebrated  optician  of  Munich  who 
first  studied  and  gave  a  detailed  description  of  them. 
They  represent  missing  images  of  the  slit,  and  indicate  that 
the  light  which  would  occupy  these  positions  in  the  com- 
plete spectrum  is  missing  from  sunlight.  When  the  slit 
is  a  little  wider  or  the  lens  slightly  out  of  focus,  the  dark 


DISPERSION  OF  LIGHT.     COLOR  441 

lines  are  obliterated  by  the  overlapping  of  the  adjacent 
images.  (A  detailed  account  of  the  Fraunhofer  lines  is 
given  in  Arts.  492-495.) 

355.  The  Nature  of  Color.  —  When  we  call  one  part 
of  the  spectrum  red,  another  part  green,  etc.,  we  are  only 
naming  the  color  sensations  which  we  experience  when  these 
different  lights  stimulate  the  optic  nerve.  The  character 
of  the  sensation  is  determined  by  the  wave  length  of  the 
light.  In  fact,  color,  as  a  property  of  the  light  itself,  is 
nothing  else  than  its  wave  length  or  the  frequency  of  the 
ether  vibrations.  The  sensation  of -color  bears  the  same 
relation  to  the  wave  length  of  light  that  the  sensation  of 
pitch  does  to  the  length  of  sound  waves. 

The  measurement  of  the  wave  lengths  of  light  is  based 
on  principles  and  methods  which  belong  to  advanced 
physics.  By  such  measurements  it  is  found  that  red  light 
consists  of  the  longest  visible  ether  waves,  and  violet  of 
the  shortest.  In  the  complete  spectrum  all  possible  wave 
lengths  between  these  limits  are  present,  decreasing  in  an 
unbroken  series  from  the  extreme  red  to  the  extreme  vio- 
let. Corresponding  to  these  innumerable  wave  lengths, 
there  is  a  continuous  gradation  of  color  from  one  end  of 
the  spectrum  to  the  other.  The  most  dissimilar  colors  are 
violet,  blue,  green,  yellow,  and  red.  To  the  normal  eye  these 
bear  no  resemblance  to  one  another.  In  addition  to  these 
we  recognize  the  intermediate  colors,  violet-blue  or  indigo, 
between  the  violet  and  the  blue;  greenish  blue  and  bluish 
green,  between  the  blue  and  the  green;  yellowish  green 
and  greenish  yellow,  between  the  green  and  the  yellow; 
and  yellowish  red,  or  orange,  between  the  yellow  and 
the  red. 

As  the  temperature  of  a  body  rises,  the  vibrations  of 


442  LIGHT 

its  molecules  become  more  and  more  rapid,  and  shorter 
waves  are  set  up  in  the  ether.  At  about  525°  C.  some  of 
the  molecules  vibrate  with  sufficient  rapidity  to  give  out 
red  light.  As  the  temperature  continues  to  rise,  additional 
colors  are  given  out  in  order  from  red  to  violet,  and  the 
color  of  the  body  changes  from  red  through  orange  and  yel- 
low to  white.  Sunlight,  and  the  light  from  white-hot 
bodies  in  general,  is  a  mixture  of  all  or  practically  all  wave 
lengths  within  the  range  of  visibility.  Such  light  may 
be  compared  to  an  utterly  discordant  chaos  of  sound  of 
every  audible  pitch. 

356.  The  Cause  of  Dispersion.  —  The  dispersion  of  light 
is  the  result  of  unequal  refraction  of  its  constituent  colors. 
As  -shown  in  the  figures,  the  violet  or  shortest  waves  are 
refracted  most,  and  the  red  or  longest  waves  least.  Accord- 
ing to  the  wave  theory,  this  can  only  be  due  to  a  greater 
retardation  of  the  shorter  waves  in  passing  through  a  re- 
fractive medium.  In  a  vacuum  waves  of  all  lengths  travel 
with  the  same  velocity. 

367.  Invisible  Portions  of  the  Spectrum.  —  The  wave  lengths 
of  light  vary  from  .00077  mm-  for  the  extreme  red  to  .00039  mm. 
for  the  extreme  violet.  The  interval  between  these  extremes, 
expressed  as  in  music,  is  slightly  less  than  one  octave.  (Read  again 
Art.  297.) 

The  solar  spectrum  includes  two  octaves  of  ether  waves  beyond 
the  violet  end,  and  more  than  six  octaves  beyond  the  red  end.  The 
part  beyond  the  violet  is  called  the  ultra-violet  spectrum.  Although 
invisible,  it  can  be  photographed.  In  fact,  the  photographic  plate 
is  more  sensitive  to  ultra-violet  waves  than  it  is  to  the  visible  spectrum. 
The  part  of  the  spectrum  below  the  red  is  called  the  infra-red  spectrum. 
Only  a  small  portion  of  it,  at  its  upper  end,  has  ever  been  photo- 
graphed. It  has  been  studied  mainly  by  its  heating  effect.  For 
this  purpose  the  late  Professor  Langley  invented  an  instrument  by 


DISPERSION  OF  LIGHT.     COLOR  443 

which  he  could  detect  a  variation  in  temperature  of  one  hundred- 
millionth  of  a  Centigrade  degree. 

The  distribution  of  energy  in  the  solar  spectrum  has  been  made  the 
subject  of  careful  investigation.  It  is  determined  by  measuring 
the  heating  effect  of  the  waves,  when  absorbed  by  a  black  surface. 
The  region  of  maximum  energy  is  found  to  be  in  the  yellow  and  green; 
but  the  energy  of  all  the  infra-red  waves,  taken  together,  greatly 
exceeds  that  of  the  visible  spectrum.  The  blue  waves  possess  very 
little  energy,  the  violet  and  the  ultra-violet  still  less.  In  the  spectra 
of  even  the  most  efficient  artificial  sources  of  light,  such  as  the  electric 
arc,  the  region  of  maximum  energy  lies  far  below  the  red;  and  the 
energy  of  all  the  luminous  waves  together  is  only  a  very  small  per- 
centage of  the  whole  radiation.  The  great  problem  of  artificial  illu- 


FIG.  335.  —  Chromatic  Aberration. 

mination  is  to  discover  some  means  of  producing  a  good  white  light 
which  represents  a  reasonable  fraction  of  the  energy  expended  in 
producing  it.  Great  advance  has  been  made  in  this  direction  in 
recent  years,  especially  in  the  newer  forms  of  electric  lamps. 

Much  longer  ether  waves  than  those  of  the  infra-red  radiation 
from  hot  bodies  can  be  produced  by  electrical  means.  These  vary 
in  length  from  a  few  millimeters  to  many  meters.  It  is  such  waves 
that  are  made  use  of  in  wireless  telegraphy. 

358.   Chromatic  Aberration.     Achromatic  Lenses.  —  In 

studying  lenses  and  their   use   in   optical 

instruments   we   had    occasion    to    note   a 

certain    imperfection    of    focusing,    called 

chromatic  aberration  (Arts.  335,  345,  346,   FlG-  336.— Achro- 

and  347).    The  nature  of  this  imperfection 

is   shown   in    Fig.    335.     The   constituent   colors   of   the 

incident  light  are  refracted   unequally,  violet  most  and 


444 


LIGHT 


red  least,  just  as  with  prisms.     Hence  the  violet  waves 
of  white  light  from  a  point  source,  S,  are  brought  to  a 
A  focus  at  one  point,  v,  and  the  red 

waves  to  a  more  distant  focus,  r, 
while  the  other  colors  are  focused  at 
intermediate  points.  When  a  screen 
is  placed  at  r,  we  see  a  red  image  of 
the  point  surrounded  by  a  circle  of 
blue  and  violet  light;  when  the  screen 
is  at  v,  we  see  a  blue-violet  image 
surrounded  by  orange  and  red. 

The  dispersion  of   the  colors  in- 
PIG.  337.  — Action  of  the    creases  from  the  central  portion  of 

greater  refraction  and  greater  dispersion  go  together. 
Hence  a  diaphragm,  with  a  small  circular  opening  for  the 
central  rays,  serves  the  double  purpose  of  diminishing  both 
spherical  and  chromatic  aberration.  But  this  is  only  a 
partial  remedy  at  the  best,  and  it  will  not  serve  at  all 
where  a  large  amount  of  light  is  re- 
quired. 

A  century  and  a  half  elapsed  after  the 
telescope  and  the  microscope  were  in- 
vented before  it  was  discovered  that 
chromatic  aberration  could  be  corrected 
by  combining  a  convex  lens  of  crown 
glass  with  a  concave  lens  of  flint  glass 
(Fig.  336).  The  refractive  power  of 
flint  glass  is  only  slightly  greater  than 
that  of  crown,  while  its  dispersive  power 
is  nearly  twice  as  great.  Hence  if  the  focal  length  of  the 
concave  lens  is  about  twice  that  of  the  convex  lens,  it  will 
produce  an  equal  and  opposite  dispersion,  with  only  half  the 


FIG.     338.  — High- 
grade  Camera  Lens. 


DISPERSION  OF  LIGHT.     COLOR  445 

deviation  (Fig.  337,  A  and  B),  and  the  light  after  passing 
through  both  will  still  be  convergent,  but  not  dispersed 
(Fig.  337C).  Double  lenses  constructed  on  this  prin- 
ciple are  called  achromatic  lenses.  They  are  used  as 
objectives  in  telescopes,  opera  glasses,  and  microscopes, 
also  in  projection  lanterns,  cameras,  etc.  Achromatic 
eyepieces  are  somewhat  differently  constructed,  but  accom- 
plish the  same  result. 

Chromatic  aberration  can  not  be  wholly  avoided  with  two  lenses 
only.  Some  of  the  colored  rays  are  brought  exactly  together  again, 
but  not  all.  The  best  objectives  for  lanterns  cameras  (Fig.  338), 
and  microscopes  (Fig.  339)  have  three  or  more  lenses.  Such  com- 
binations are  not  only  perfectly  achromatic,  but  are  free  from  other 
imperfections  of  focusing  as  well. 

359.  The  Rainbow  is  a  solar  spectrum,  formed  by  the 
dispersion  of  sunlight  by  drops  of  water  in  falling  rain, 
and  in  the  spray  of  fountains,  waterfalls,  etc.  Sometimes 
one  bow  is  seen,  sometimes  two.  They 
are  always  arcs  of  circles;  and,  when 
two  are  formed,  they  are  concentric 
(Fig.  340).  The  inner  or  lower  one 
is  called  the  primary  bow,  and  the  other 
the  secondary  bow.  The  primary  bow 
is  always  much  the  brighter.  In  it  the 
red  is  on  the  outside,  the  violet  on  the 
inside.  In  the  secondary  bow  the  order 
of  the  colors  is  reversed.  Rainbows 

•    *  •      ,          ».          .  FIG.  339. — Microscope 

are  always  seen  in  the  direction  op-  objective,  One-Sixth 
posite  to  the  sun,  with  the  sun,  the  Inch- 
observer,  and  the  center  of  the  circular  arc  in  the  same 
straight  line,  EO.  This  line  is  called  the  axis  of  the  bow. 
The  action  of  the  individual  drops  in  forming  a  rain- 
bow can  be  shown  on  a  large  scale,  with  the  aid  of 


446 


LIGHT 


a  slender  sunbeam  in  a  darkened  room  and  a  globe 
filled  with  water.  A  round-bottomed  flask  will  answer 
the  purpose  very  well.  When  the  globe  is  held  in  the 

path  of  the  beam,  the  greater  part 
of  the  light  passes  through  it;  the 
remainder  undergoes  one  or  more 
internal  reflections  before  it  is 
refracted  out.  The  primary  bow 
is  formed  by  light  that  is  reflected 
once  (Fig.  341),  when  the  angle  of 
incidence  at  A  is  approximately 

FIG.  340.  —  Primary   and   Sec-    59°.      With  this  adjustment  of  the 

globe  and  a  white  screen,  XY,  in 


ondary  Rainbows. 


the  path  of  the  emergent  light,  a  curved  spectrum  is  seen, 

with  the  violet  at  its  inner  edge.     Part  of  the  dispersion  is 

due  to  each  refraction,  as  shown  in  the  figure.    When  the 

beam  is  incident  less  obliquely,  the  different  colors  are 

spread  out,  without  separation, 

as  a  broad  band  of  faint  white 

light,    and     no     spectrum     is 

formed.     When    the   angle    of 

incidence  is  slightly  greater,  as 

at  A  in  Fig.  342,  the  light  that 

is     twice    internally    reflected 

forms  the  secondary  spectrum,    y 

This  spectrum  is  faint,  owing 

to  the  additional  loss  of  light 

at  the  second  reflection.    With 

an  incident  beam  large  enough 

to  cover  the  globe,  the  primary  spectrum  forms  a  complete 

spectrum  on  the  screen.     In  the  strong  illumination  the 

secondary  spectrum  is  very  faint,  if  not  invisible. 

In  looking  at  a  rainbow  (Fig.  340),  the  eye  receives  only 


FIG.     341.  —  Dispersion    in    the 
Primary   Rainbow. 


DISPERSION  OF  LIGHT.    COLOR  447 

a  single  color  from  any  one  drop;   and  this  color  is  the 

same  for  all  drops  which  are  at  the  same  angular  distance 

from    the    axis    of    the    bow. 

Hence    the    bow    is    circular. 

The  inner  edge  of  the  primary 

bow  is  at  an  angle  of  40°  from     54°  51' 

the  axis  and  its  outer  edge  at 

42°.    The  edges  of  the  secondary 

bow  are  at  angles  of  51°  and  54° 

respectively. 

At  sunrise  or  sunset  a  rain- 
bow, if  complete,  appears  as  FlG' 
a  semicircle  (Fig.  340),  its  axis 
being  horizontal.  Since  the  center  of  the  bow  is  always 
at  the  same  angle  below  the  horizon  that  the  sun  is  above 
it,  the  higher  the  sun  is  the  shorter  will  be  the  arc  of  the 
bow.  When  the  sun  is  more  than  42°  above  the  horizon, 
only  the  secondary  bow  can  be  seen. 

360.  Color  of  Bodies.  —  What  we  commonly  regard  as 
the  natural  color  of  an  object  is  really  the  color  of  the  light 
that  the  object  transmits  or  reflects  when  white  light  falls 
upon  it.  A  body  that  is  transparent  to  light  of  all  wave 
lengths  is  colorless,  e.g.  window  glass  and  water.  A  body 
that  reflects  light  of  all  wave  lengths  in  equal  proportions 
is  white  if  it  has  a  high  reflecting  power,  gray  if  its  reflecting 
power  is  rather  low,  and  black  if  it  reflects  almost  no  light. 
A  body  absorbs  the  incident  light  that  it  does  not  transmit 
or  reflect.  If  it  transmits  none,  it  is  opaque;  if  it  reflects 
practically  none,  it  is  black. 

If  a  body  transmits  waves  of  different  lengths  in  unequal 
proportions,  or  transmits  some  and  wholly  absorbs  others, 
the  transmitted  light  is  colored,  and  its  color  is  called  the 


448  LIGHT 

color  of  the  body.  Colored  glass  and  colored  liquids  of 
various  kinds  are  familiar  examples.  A  colored  opaque 
body  owes  its  color  to  the  fact  that  it  reflects  waves  of  dif- 
ferent lengths  in  unequal  proportions,  or  else  reflects  some 
and  wholly  absorbs  others. 

Whatever  the  color  of  a  body,  the  analysis  of  the  light 
transmitted  or  reflected  by  it  shows  that  this  light  is  com- 
posed of  certain  spectral  colors  (colors  of  the  spectrum)  in 
certain  porportions.  There  are  no  simple,  indivisible,  or 
elementary  colors  other  than  those  of  the  complete  spectrum; 
and  any  composite  color  can  be  analyzed  into  its  elementary 
or  spectral  constituents  by  means  of  a  prism.  This  analy- 
sis gives  a  spectrum  in  which  the  constituent  parts  are 
arranged  as  in  the  solar  spectrum,  the  only  difference  being 
that  certain  parts  of  the  complete  spectrum  are  missing 
and  others,  perhaps,  relatively  weak.  In  analyzing  the 
light  from  any  body  we  may  make  use  of  either  the  real 
or  the  virtual  spectrum.  The  study  of  virtual  spectra 
is  adapted  to  individual  laboratory  work  (see  Lab.  Ex. 
56).  For  class  observation  the  spectrum  is  projected  on 
a  screen  in  a  darkened  room,  as  in  the  following  experi- 
ments. 

Let  a  pure  solar  spectrum  be  projected  on  a  screen,  as 
in  Art  354,  and  one  end  of  the  slit  covered  with  a  piece 
of  red  (ruby)  glass.  The  solar  spectrum  and  the  spectrum 
of  the  light  transmitted  by  the  glass  will  appear  upon  the 
screen,  one  above  the  other.  The  latter  consists  of  red, 
with  perhaps  a  little  orange.  The  other  constituents  of 
white  light  are  absorbed  by  the  glass.  Substituting  blue 
(cobalt)  glass,  the  spectrum  of  the  transmitted  light  will 
be  found  to  consist  of  violet,  blue,  green,  and  a  little  red. 
Yellow  glass  transmits  red,  orange,  yellow,  and  green. 
Colored  solutions  of  chemicals,  such  as  copper  sulphate 


DISPERSION  OF  LIGHT.    COLOR  449 

and  potassium  bichromate,  can  be  tested  in  the  same  way. 
A  flat  bottle  will  serve  for  holding  the  liquid/but  a  narrow 
tank  of  plate  glass  is  better. 

When  two  transparent  bodies  of  different  color  are 
placed  before  the  slit,  one  in  front  of  the  other,  the  first 
absorbs  certain  constituents  of  the  incident  light,  the 
second  absorbs  certain  other  constituents;  and  the  spectrum 
of  the  transmitted  light  consists  only  of  the  color  or  colors 
which  are  common  to  the  light  transmitted  by  the  two 
separately.  Thus  green  is  the  only  spectral  color  trans- 
mitted by  either  blue  or  yellow  glass  that  is  also  trans- 
mitted by  the  other ;  hence  the  two  together  appear  green. 
Similarly  the  combination  of  red  and  blue,  red  and  green, 
or  orange  and  blue  glass  is  very  nearly  opaque,  since  no 
color  that  they  separately  transmit  in  considerable  quantity 
is  common  to  both. 

The  light  reflected  by  a  colored  opaque  body  can  be 
analyzed  in  a  very  interesting  way  by  noting  the  appear- 
ance of  the  body  when  it  is  held  successively  in  the 
different  colors  of  a  large  solar  spectrum,  projected  on  a 
screen.  A  white  card  held  in  the  violet  light  appears  to 
be  violet;  in  the  blue  it  appears  blue;  in  the  green,  green, 
etc.  Whatever  the  color  of  the  light  by  which  it  is 
illuminated,  it  appears  to  be  of  that  color.  In  white 
light  it  appears  of  its  natural  color  —  white  —  for  it  re- 
flects all  the  constituents  of  white  light  in  equal  pro- 
portions, and  absorbs  but  little  of  any.  A  piece  of  green 
paper  will  appear  black  in  the  violet,  indigo,  orange,  or 
red;  in  the  blue  it  will  probably  appear  to  be  a  dark 
blue,  and  in  the  yellow  a  dirty  yellow,  due  to  the  reflec- 
tion of  a  little  of  these  colors;  in  the  green  it  will  appear 
at  least  very  nearly  of  its  natural  color.  Similarly  we 
can  determine  the  spectral  colors  that  any  colored  body 


450  LIGHT 

is  capable  of  reflecting;  and  these  will  be  the  constitu- 
ents of  the  light  that  it  reflects  when  it  is  illuminated  by 
white  light. 

Most  artificial  lights  are  deficient  in  violet  and  blue,  and 
hence  are  more  or  less  yellowish.  In  such  a  light,  white 
has  the  appearance  of  pale  yellow,  and  blue  is  often  mis- 
taken for  green.  The  greenish  appearance  of  blue  is  due 
to  the  fact  that  blue  pigments  reflect  violet  and  green  as 
well  as  blue  light,  and  green  predominates  in  the  light  that 
they  reflect  when  the  incident  light  is  weak  in  the  violet 
and  blue. 

The  light  transmitted  or  reflected  by  colored  bodies  in  general 
is  composite,  consisting  in  many  cases  of  fully  half  of  the  complete 
spectrum.  The  composite  character  of  light  can  not  be  detected  by 
the  unaided  eye;  for  the  eye  is  absolutely  wanting  in  the  power  of 
analysis.  If  the  ear  were  similarly  deficient,  we  could  not  distinguish 
the  constituents  of  a  complex  sound.  The  notes  sounded  simul- 
taneously by  an  orchestra  would  produce  the  sensation  of  a  single 
note  of  average  pitch,  and  harmony  and  discord  would  alike  be 
unknown. 

361.  Colors  of  the  Sky.  —  A  gas  or  a  liquid  which,  of  itself,  is 
colorless  becomes  colored  when  it  contains  finely  divided  matter  in 
suspension.  An  excellent  example  is  the  sky-blue  liquid  obtained 
by  adding  to  water  a  very  small  proportion  of  milk  or  an  alcoholic 
solution  of  mastic,  or  by  mixing  a  few  drops  of  dilute  nitrate  of  silver 
with  a  quantity  of  water  in  which  a  little  table  salt  has  been  dis- 
solved. (In  the  last  case  chloride  of  silver  is  formed,  which  is  insol- 
uble in  water,  but  remains  suspended  in  the  form  of  extremely  minute 
solid  particles.  The  same  is  true  of  the  mastic.)  These  liquids 
appear  blue  by  reflected  light;  but  are  yellow  or  orange  when  viewed 
by  transmitted  light.  This  is  due  to  the  fact  that  the  suspended 
particles  reflect  a  considerable  part  of  the  violet  and  blue  light,  but 
reflect  less  and  less  of  the  other  colors  toward  the  red  end  of  the  spec- 
trum. Thus  violet  and  blue  predominate  in  the  reflected  light,  and 
red,  orange,  and  yellow  in  the  transmitted  light. 

The  blue  color  of  the  sky  is  similarly  explained,  the  air  being  ren- 


DISPERSION  OF  LIGHT.     COLOR  451 

dered  visible  against  the  dark  background  of  black  space  by  sunlight 
reflected  from  its  fine  suspended  dust  or  water  particles;  while  the 
light  transmitted  directly  from  the  sun  is  always  more  or  less 
yellowish,  and,  in  the  afternoon  and  evening,  when  sunlight  comes 
to  us  through  a  greater  thickness  of  the  dusty  layers,  verges  toward 
orange  or  even  red. 

362.   Color    Sensation.     Complementary    Colors.  —  We 

have  seen  that  the  same  color  sensation  may  be  produced 
by  light  of  one  wave  length,  selected  from  the  spectrum,  or 
by  composite  light  of  many  different  wave  lengths.  Thus 
the  light  transmitted  by  yellow  glass  may  appear  to  the 
eye  exactly  like  the  yellow  of  the  spectrum,  although,  when 
analyzed,  it  is  found  to  consist  of  red,  orange,  yellow, 
and  green  waves.  More  curi- 

.  .„  .  Purple 

ously  still,  a  given  color  sensa- 
tion may  be  due  to  composite 
light  in  which  the  wave  length 
corresponding  to  that  color  is 
wholly  wanting.  For  example, 
a  mixture  of  spectral  red  and 
spectral  yellow  produces  the 
sensation  of  orange,  and  a  mix- 
ture of  spectral  violet  and  green 
produces  the  sensation  of  blue.  FlG'  343' "  °pposite  Colors  Com- 

plementary. 

In  general  the  mixture  of  any 

two  spectral  colors  appears  to  the    eye   to  be  identical 

with   the   color   named    midway    between    them    in   the 

accompanying    chart    (Fig.    343).      The   mixture    of    red 

and  violet  lights  is  purple,  —  a  color  not  found  in  the 

spectrum. 

The  mixture  of  any  pair  of  spectral  colors  named  on 
opposite  sides  of  the  chart  appears  white,  when  the  two 
lights  are  taken  in  the  right  proportion.  With  the  un- 


452  LIGHT 

aided  eye  these  different  white  lights  can  not  be  dis- 
tinguished from  one  another  or  from  ordinary  white  light, 
which  is  a  mixture  of  all  the  spectral  colors;  but  the  prism 
would  instantly  reveal  their  differences.  Any  two  colored 
lights  which  together  produce  the  sensation  of  white  are 
called  complementary  colors. 

All  possible  color  sensations,  including  white,  can  be 
produced  by  combining  spectral  red,  green,  and  violet 
lights  in  different  proportions.  Red,  green,  and  violet  are 
therefore  called  the  primary  color  sensations. 

In  studying  mixtures  of  colored  lights,  the  selected  colors  of  the 
spectrum  can  be  focused  by  a  lens  or  reflected  by  mirrors  to  the  same 
spot  on  a  white  screen.  The  same  effect  is  more  conveniently  pro- 
duced by  means  of  colored  disks,  each  of  which  is  slit  along  a  radius, 
thus  permitting  any  desired  amount  of  overlapping  when  two  or  more 
of  the  disks  are  placed  together  on  an  axis  through  their  common 
center  (Fig.  344).  When  the  disks  are  rapidly  rotated  about  the  axis, 
only  one  color  is  seen,  and  this  covers  the  entire  circular  area.  This 
result  is  due  to  the  fact  that  the  sensation  of  sight  continues  for  a 
fraction  of  a  second  after  the  light  ceases  to  enter  the  eye  or  ceases 
to  fall  upon  the  same  part  of  the  retina.  The 
rapid  rotation  causes  the  different  colors  to 
come  from  all  parts  of  the  disk  in  such  rapid 
succession  that  each  color  produces  a  con- 
tinuous impression  for  the  entire  circular 
area,  just  as  if  it  were  reflected  by  the  en- 
tire area.  The  colors  of  the  disk  are  com- 
posite;  but  their  effect  in  a  mixture  is  the 
same  as  if  they  were  simple  spectral  colors. 
In  most  cases,  however,  the  total  amount  of  reflected  light  is 
so  small  that  the  resultant  color  is  deficient  in  brightness.  Com- 
plementary colors,  for  example,  generally  yield  a  dark  gray  instead 
of  white. 

363.  The  Young-Helmholtz  Theory  of  Color  Sensation.  Color 
Blindness.  —  The  most  probable  theory  of  color  sensation  is  that 
proposed  by  the  English  physicist,  Dr.  Thomas  Young,  and  further 


DISPERSION  OF  LIGHT.     COLOR  453 

amplified  by  Helmholtz.  .According  to  this  theory  the  normal  eye 
is  provided  with  three  sets  of  nerves,  which  are  most  sensitive  to  red, 
green,  and  blue-violet  light  respectively.  A  primary  color  sensa- 
tion is  due  mainly,  if  not  wholly,  to  the  stimulation  of  one  set  of  nerves. 
All  other  color  sensations  are  resultant  effects,  due  to  the  stimula- 
tion of  two  or  of  all  the  sets  in  different  degrees.  For  example,  the 
sensation  of  yellow  is  produced  when  the  "red"  and  the  "green" 
nerves  are  equally  stimulated,  whether  by  a  mixture  of  red  and  green 
lights  or  by  spectral  yellow. 

A  person  who  does  not  see  all  the  colors  of  the  spectrum  as  they 
appear  to  the  normal  eye  is  said  to  be  color  blind.  Usually  it  is  the 
red  that  is  seen  abnormally;  in  rare  instances  it  is  the  green  or 
the  violet.  This  defect  of  vision  is  supposed  to  be  due  to  the  absence 
or  inactivity  of  the  corresponding  set  of  nerves.  In  red  blindness  red 
is  perceived  by  a  weak  stimulation  of  the  "green"  nerves,  and  it 
is  distinguished  from  green  only  as  a  darker  shade  of  the  same  color. 
The  absence  of  the  red  sensation  modifies  the  other  color  sensations 
more  or  less,  with  the  probable  exception  of  blue  and  violet. 

Extensive  tests  have  shown  that  three  or  four  per  cent,  of  all  per- 
sons are  color  blind.  The  defect  usually  exists  from  birth,  and 
doubtless  in  most  cases  is  never  discovered.  For  the  person  thus 
afflicted  learns  in  childhood  to  call  colors  by  their  right  names,  with- 
out having  any  reason  to  suspect  that  his  color  sensations  are  differ- 
ent from  those  of  his  companions.  He  might  sometimes  wonder 
why  other  children  could  spy  out  ripe  strawberries  among  the  green 
vines  more  readily  than  himself;  but  he  would  most  assuredly  not 
hit  upon  the  true  reason.  However,  a  simple  test  has  been  devised 
by  which  color  blindness  can  be  detected  with  certainty.  The  per- 
son undergoing  the  test  is  directed  to  assort  a  large  number  of  vari- 
ously colored  skeins  of  wool,  placing  together  all  that  resemble  each 
other.  The  colors  are  so  chosen  that  one  who  is  color  blind  will  be 
sure  to  make  mistakes.  Mariners,  soldiers,  and  railway  employes 
are  thus  examined  as  a  test  of  their  ability  to  distinguish  colored 
signals. 

364.  Colors  of  Mixed  Pigments.  —  A  mixture  of  blue 
and  yellow  lights  is  white;  but  the  light  transmitted  through 
blue  and  yellow  glass  in  succession  is  green.  The  results 


454  LIGHT 

are  not  inconsistent,  for  they  are  obtained  by  wholly  dif- 
ferent processes.  The  first  is  a  case  of  addition,  the  sec- 
ond a  case  of  double  subtraction,  as  already  explained. 
The  mixture  of  blue  and  yellow  paints  or  powders  is  green. 
This  is  also  a  case  of  double  subtraction.  The  blue  paint 
absorbs  the  red,  orange,  and  yellow  of  the  incident  light, 
and  the  yellow  paint  absorbs  the  violet  and  blue.  Green 
is  the  only  color  not  strongly  absorbed  by  one  or  the  other, 
and  hence  is  the  principal  color  in  the  light  reflected  by  the 
mixture.  In  general,  the  light  reflected  by  mixed  pigments 
consists  of  the  colors  which  are  not  absorbed  by  any  of 
the  constituents.  If  the  light  reflected  by  one  pigment 
has  no  constituent  in  common  with  the  light  reflected 
by  a  second  pigment,  a  mixture  of  the  two  is  black. 
This  is  the  case  with  vermilion  (a  bright  red)  and  ultra- 
marine (a  deep  blue). 

The  artist  mixes  his  pigments  before  applying  them  to  the 
canvas.  In  making  colored  prints  the  different  pigments  are  laid 
on,  one  after  the  other,  in  separate  impressions.  The  inks  are 
differently  distributed,  so  that  the  final  color  is  in  some  places  due 
to  one  only,  in  other  places  to  two,  and  in  still  others  to  all  three  in 
varying  proportions.  Thus  a  surprising  number  of  delicate  tints 
and  shades  are  produced. 

365.  Color  by  Interference.  —  White  light  becomes  colored 
whenever  it  loses  one  or  more  of  its  constituents,  whether  by 
selective  absorption  or  any  other  process.  The  rainbow  colors 
of  soap  bubbles  and  of  thin  films  of  oil  floating  on  water  are 
due  to  the  loss  of  certain  wave  lengths  by  interference.  The 
phenomenon  is  similar  to  the  interference  of  sound  waves  (Art. 
277),  and  takes  place  under  similar  conditions,  as  in  the  following 
experiment: 

Let  two  pieces  of  clean  plate  glass  be  pressed  firmly  together 
with  a  small  clamp,  and  held  so  that  a  strong  light  falls  upon 
them.  From  the  illuminated  side,  brilliant  colored  bands  will 


DISPERSION  OF  LIGHT.     COLOR  455 

be  seen,  surrounding  the  point  of  closest  contact.  When  pres- 
sure is  applied  at  other  points  with  the  fingers,  the  bands  become 
wider  and  shift  into  new  positions,  showing  that  the  color  varies 
with  the  distance  between  the  plates.  The 
nature  of  these  color  effects  can  be  under- 
stood, in  a  general  way,  with  the  aid  of 
Fig.  345.  MM  and  NN  are  sections  of 
the  glass  plates,  the  distance  between  them 
being  greatly  exaggerated.  Light  incident 
along  the  path  AB  is  partially  reflected  at  C  ^Jjj  Jjp 

from  the  lower  surface  of  the  upper  plate,  FlG 

and  also  at  E  from  the  upper  surface  of  the 

lower  plate.  Some  of  the  light  reflected  at  E  is  transmitted  through 
the  upper  plate,  parallel  to  and  nearly  coincident  with  the  light 
reflected  from  C.  But,  in  twice  crossing  the  space  between  the 
plates,  the  waves  reflected  at  E  fall  behind  the  corresponding 
waves  reflected  at  C,  and  waves  of  a  certain  length  in  the  two  sets 
will  meet  in  opposite  phase  and  destroy  each  other.  The  re- 
flected light  is  complementary  to  the  color  lost  by  interference; 
and  the  latter  differs  at  different  places,  depending  upon  the  distance 
between  the  plates. 

Interference  colors  are  produced  whenever  light  is  reflected  from 
the  two  surfaces  of  a  very  thin  film  or  plate  of  any  transparent  sub- 
stance. They  are  most  common  with  liquid  films,  such  as  soap 
bubbles  and  films  of  oil  on  water.  In  the  above  experiment  the  film 
is  the  thin  sheet  of  air  between  the  plates.  Solids  are  rarely  thin 
enough  to  exhibit  such  colors,  mica  in  very  thin  flakes  being  the  only 
common  example. 

Interference  also  occurs,  with  even  more  beautiful  color  effects, 
when  light  is  reflected  from  surfaces  covered  with  minute  parallel 
grooves  and  ridges,  called  striations,  as  in  mother-of-pearl  and  the 
plumage  of  many  birds.  As  such  a  surface  is  turned  about,  so  that 
the  light  falls  upon  it  at  changing  angles,  brilliant  rainbow  colors 
sweep  over  it  in  rapid  succession.  This  beautiful  "  play  of  colors  " 
is  often  seen  on  the  breasts  of  humming  birds,  as  they  dart  about  in 
the  sunshine.  Bodies  that  exhibit  interference  colors  are  said  to  be 
iridescent. 


456  LIGHT 

PROBLEMS 

1.  What  is  the  function  of  the  lens  in  producing  a  pure  spectrum? 

2.  Why  is  it  not  possible  to  correct  the  chromatic  aberration  of  a  lens  by 
any  change  in  the  form  of  its  surfaces? 

3.  (a)  State    all    the    conditions    necessary    for    a    rainbow.     (6)  Do 
two  observers  see  exactly  the  same  rainbow? 

4.  Prove  that  the  reflection  within  a  rain-drop  takes  place  at  less  than 
the  critical  angle,  and  is  therefore  not  total. 


CHAPTER   XI 
MAGNETISM 

I.   PROPERTIES  OF  MAGNETS 

366.  Natural   and   Artificial   Magnets.  —  A  magnet  is 
distinguished  from  other  bodies  by  its  power  of  attracting 
pieces  of  iron.     This  power  is  a  property  of  the  material 
composing  the  magnet,  and  may  be  either  temporary  or 
permanent.     Natural  magnets  were  known  to  the  ancients. 
They  are  black  stones,  consisting  of  a  certain  iron  ore  called 
magnetic  oxide  of  iron  or  magnetite  (FesO^.     The  word 
magnet  is  derived  from  Magnesia,  the  name  of  the  city  in 
Asia  Minor  near  which  these  magnetic  stones  were  first 
found.     After  the   discovery,  in  the  eleventh  or  twelfth 
century,  that  suspended  magnets  always  point  in  a  defi- 
nite direction,  they  came  into  use  for  determining  direc- 
tions on  land  and  sea,  and  were  called  lodestones  (leading 
stones). 

When  a  piece  of  highly  tempered  steel  is  rubbed  with  a 
magnet  or  in  any  other  way  subjected  to  strong  magnetic 
action,  it  acquires  permanent 'magnetic  proper- 
ties, and  becomes  a  manufactured  or  artificial 
magnet.  Magnets  are  made  of  various  shapes, 
which  are  adapted  to  different  uses  (Figs.  346, 
347,  and  348). 

367.  The  Poles  of  a  Magnet  — The  different  H0Grs3e4s6h~ 
parts  of  a  magnet  have  very  unequal  power  of  Magnetand 
attracting  iron,  as  is  shown  by  the  distribution  or  Arma- 
of  the  mass  of  filings  or  small  tacks  which  the  ture> 

457 


458  MAGNETISM 

magnet  can  hold  (Fig.  347).  The  quantity  is  greatest  at 
and  near  the  ends,  and  diminishes  rapidly  toward  the 

middle    portion,    which     is 
generally  bare.     The  regions 
near    the    ends    where    the 
FIG.  347.  magnetic  action  is  strongest 

are  called  the  poles  of  the  magnet. 

A  magnet  has  regularly  two  poles,  one  at  each  end, 
whatever  its  shape.  By  irregular  magnetization  additional 
poles  can  be  developed;  but  such  cases  are  unimportant 
and  need  not  be  considered. 

When  a  straight  magnet  is  suspended  or  supported  so 
that  it  is  free  to  turn  in  a  horizontal  plane,  and  is  subjected 
only  to  the  magnetic  action  v 

,     ,  ,  S        ^  N  Mn(mft^ 

of  the  earth  (as  will  be  ex- 
plained later),  it  always 
comes  to  rest  with  the  same 
end  pointing  in  a  northerly 

*^  *  FIG.  348.  —  Magnetic  Needle. 

direction.    This  end  of  the 

magnet  is  called  the  north  pole  (i.e.  the  north-seeking  pole), 
and  the  other  end  is  called  the  south  pole.  A  slender 
magnet,  balanced  on  a  pivot  (Fig.  348)  or  suspended  at  its 
center  by  an  untwisted  fiber  is  called  a  magnetic  needle. 

368.  Magnetic  Attraction  and  Repulsion.  —  When  the 
north  pole  of  a  bar  magnet  is  brought  near  the  north  pole 
of  a  magnetic  needle,  the  latter  is  driven  away,  or  repelled. 
Similarly  the  south  pole  of  the  needle  is  repelled  by  the 
south  pole  of  the  magnet.  But  the  north  pole  of  the  needle 
is  drawn  toward  the  south  pole  of  the  magnet,  and  its 
south  pole  toward  the  north  pole.  If  the  bar  magnet  is 
suspended  so  that  it  also  is  free  to  move,  or  if  the  experi- 
ment is  tried  with  two  magnetic  needles,  it  will  be  found 


PROPERTIES   OF   MAGNETS  459 

that  both  are  attracted  or  both  repelled  at  the  same  time. 
A  magnetic  force  is  always  a  mutual  action  between  two 
bodies,  in  accordance  with  Newton's  third  law  of  motion 
(Art.  118). 

Such  experiments  show  that  there  is  a  real  difference 
between  north  poles  and  south  poles,  and  that  like  poles 
repel  and  unlike  poles  attract  each  other. 

369.   Force  Exerted  between  Two  Magnetic  Poles.  —  It 

is  a  well  known  fact  that  magnets  differ  in  strength  or 
attracting  power.  Greater  strength  in  any  given  case 
may  be  due  either  to  greater  size,  or  to  a  greater  degree  of 
magnetization,  or  to  both  causes  together. 

The  force  exerted  between  two  poles  varies  as  the  prod- 
uct of  the  strengths  of  the  poles,  and  also  varies  with  the 
distance  between  them.  If  the  poles  are  small  compared 
with  the  distance,  the  attraction  or  repulsion  varies  inversely 
as  the  square  of  the  distance.  With  ordinary  magnets,  how- 
ever, this  relation  does  not  hold  very  closely.  The  effect 
of  distance  is  seen  in  the  tendency  of  two  magnets  to  move 
bodily  toward  each  other  when  their  nearer  poles  are  un- 
like, and  away  from  each  other  when  these  poles  are  alike. 
In  the  first  case  the  attraction  between  the  adjacent  poles 
exceeds  the  repulsion  of  the  more  distant  ones;  in  the 
second  case  the  repulsion  exceeds  the  attraction,  for  the 
same  reason. 

The  strengths  of  magnetic  poles  and  the  forces  which  they  exert 
upon  one  another  are  measurable  quantities;  but  their  measurement 
is  unnecessary  in  elementary  physics.  We  are  concerned,  only  in 
a  general  way,  with  their  relative  magnitudes.  Greater  or  less  mag- 
netic attraction  is  shown  by  the  more  or  less  rapid  vibration  of  a  mag- 
netic needle.  Thus  when  a  pole  of  a  magnet  is  brought  slowly  toward 
a  magnetic  needle,  the  unlike  pole  of  the  needle  swings  round  and 
vibrates  before  it,  more  and  more  rapidly  as  the  distance  decreases. 


460  MAGNETISM 

370.  Magnetic  and  Non-magnetic  Substances.  —  A  sub- 
stance which  is  attracted  by  a  magnet  and  which  can  itself 
be  magnetized  is  said  to  be  magnetic.     Substances  which  do 
not  possess  these  properties  in  any  appreciable  degree  are 
usually   classed   as   non-magnetic.     Iron   in   its   different 
forms,  such  as  cast  iron,  wrought  iron,  and  steel,  is  the 
most   strongly  magnetic   material   known.     The  next  in 
order  are  cobalt  and  nickel.     These  are  also  quite  strongly 
magnetic,  but  much  less  so  than  iron.     All  other  substances 
are  practically  non-magnetic. 

The  magnetic  properties  of  iron  are  of  very  great  impor- 
tance, being  usefully  applied  in  the  telegraph,  the  tele- 
phone, the  dynamo,  the  motor,  and  many  other  electrical 
machines  and  instruments. 

371.  Magnetic  Induction.  —  The  law  of  magnetic  action 
between  the  poles  of  two  magnets  does  not  seem  to  hold 
for  the  attraction  between  a  magnet  and  an  unmagnetized 
piece  of  iron;  for  unmagnetized  iron  has  no  poles.     Let  us 
see  what  further  may  be  learned  concerning  this  apparent 
exception. 

In  the  first  place  we  can  determine  whether  a  piece  of 
iron  is  magnetized  by  trying  to  pick  up  iron  filings  with  it. 
If  none  cling  to  it,  it  is  not  appreciably  magnetized.  A  rod 
of  soft  iron,  tested  in  this  manner,  will  be  found  to  be  un- 
magnetized. Let  the  test  be  re- 
Is  ~N]  pea  ted  at  one  end  of  the  rod,  while 
a  pole  of  a  magnet  is  held  against 
or  very  near  the  other  end.  It 
now  gathers  a  considerable  tuft  of 
filings  (Fig.  349),  which  drop  off 
as  soon  as  the  magnet  is  removed. 

FIG.  349.  — Temporary  Mag-  .  .  ,    .      .        ir 

netic  induction.  Evidently  the  rod  is  itself  a  magnet 


PROPERTIES   OF  MAGNETS  461 

while  the  permanent  magnet  is  near  it,  and  ceases  to  be 
one  when  the  magnet  is  removed.  This  action  by  which 
iron  or  steel  becomes  magnetized,  when  subjected  to 
magnetic  forces,  is  called  magnetic  induction.  In  the  case 
considered  the  induced  magnetism  is  only  temporary. 

The  north  and  south  poles  induced  in  the  iron  rod  can 
be  determined  by  means  of  a  magnetic  needle.  Thus  when 
the  north  pole  of  the  magnet  is  in  contact  with  the  rod  at 
one  end,  the  other -end  repels  the  north  pole  of  the  needle, 
and  hence  must  itself  be  a  north  pole.  The  end  in  contact 
with  the  magnet  is  then  a  south  pole,  as  shown  in  the  fig- 
ure. Hence  we  find  that  the  attraction  of  the  magnet  for 
the  supposed  unmagnetized  rod  is  really  an  attraction 
between  the  unlike  poles  of  a  permanent  magnet  and  a 
temporary  one. 

In  all  cases  where  unmagnetized  iron  or  steel  is  brought 
near  enough  to  a  magnet  to  be  attracted  by  it,  the  attrac- 
tion is  the  result  of  magnetic  induction. 

372.  Temporary  and  Permanent  Magnets.  —  When  soft 
iron,  hard  iron,  and  tempered  steel  are  subjected  to  equal 
inductive  action,  e.g.  by  contact  with  the  same  magnet,  a 
test  with  iron  filings  will  show  that  the  soft  iron  becomes 
most  strongly  magnetized  and  the  tempered  steel  the  least. 
But  the  soft  iron  loses  its  magnetism  almost  completely, 
as  soon  as  it  is  removed  from  the  influence  of  the  magnet, 
while  the  tempered  steel  retains  its  magnetism  indefinitely. 
Thus  by  magnetic  induction  a  piece  of  soft  iron  becomes 
a  temporary  magnet,  and  tempered  steel  a  permanent  one. 
Hard  iron  and  untempered  steel  retain  a  considerable  part 
of  their  induced  magnetism,  and  are  called  subpermanent. 

A  piece  of  highly  tempered  steel  can  be  made  a  permanent  magnet 
by  rubbing  it  repeatedly  from  end  to  end,  with  another  magnet,  or 


462  MAGNETISM 

from  the  center  toward  both  ends  with  unlike  poles  of  two  magnets. 
A  more  effective  method,  depending  on  the  use  of  an  electric  cur- 
rent, will  be  described  later.  Permanent  magnets  can  be  demag- 
netized or  magnetized  with  opposite  polarity,  by  sufficiently  strong 
inductive  action  in  the  opposite  direction.  Soft  iron,  as  a  magnetic 
material,  has  more  numerous  and  more  important  uses  than  tempered 
steel.  It  plays  a  necessary  part  in  the  generation  of  electric  currents 
by  dynamos,  and  also  in  a  great  many  applications  of  electricity. 

373.  Magnetic   Action   through    Bodies.     Permeability.  —  Mag- 
netic action  takes  place  through  non-magnetic  bodies  without  hin- 
drance or  modification  of  any  sort.     For  example,  a  magnet  attracts 
or  repels  a  magnetic  needle  through  a  board,  a  book,  or  a  plate  of 
glass,  just  as  if  nothing  intervened.      But  when  a  sheet  of  iron  is 
thrust  between  them,  the  needle  is  only  slightly  affected  by  the 
presence  of  the  magnet,  if  at  all.    The  sheet  of  iron,  especially  if 
large,  acts  as  a  screen  to  cut  off  magnetic  action  from  the  side  oppo- 
site to  the  magnet.     This  effect  is  due  to  induction  in  the  iron,  by 
which  it  becomes  magnetized;  and  the  mange  tic  action  is  carried  off 
to  the  edges  of  the  sheet.     This  can  be  shown  by  bringing  the  needle 
up  to  the  edge,  where  it  will  be  attracted  or  repelled  as  by  an  ordi- 
nary magnet.     A  rod  of  soft  iron,  placed  lengthwise  between  the  mag- 
net and  the  needle,  intens'fies  the  action  of  the  needle,  just  as  if  the 
magnet  had  been  brought  up  closer  to  it.     The  rod,  by  induction, 
serves  as  a  carrier  of  the  magnetic  action. 

Only  magnetic  substances  can  thus  deflect,  extend,  and  intensify 
the  action  of  a  magnet.  It  is  as  if  the  magnetic  forces  found  an 
easier  path  through  the  magnetic  substance  than  that  afforded  by  the 
air  or  other  non-magnetic  substance;  and  the  material  which  affords 
the  better  path  is  said  to  have  greater  magnetic  permeability. 

The  permeability  of  air  and  other  non-magnetic  substances  is 
taken  as  unity.  Magnetic  forces  act  with  equal  intensity  through 
all  of  them.  Nickel  and  cobalt  are  highly  permeable,  steel  is  much 
more  so,  and  soft  iron  most  of  all.  The  greater  the  permeability  of 
a  substance  the  greater  will  be  the  magnetic  induction  in  it,  when 
subjected  to  a  given  magnetizing  force. 

374.  Magnetism  is  a  Molecular  Property.  —  The  same 
magnet  may  be  used  to  magnetize  any  number  of  pieces  of 


PROPERTIES  OF  MAGNETS  463 

steel,  without  itself  becoming  weaker.  Evidently,  there- 
fore, the  induced  magnetism  is  not  something  transmitted 
from  the  magnet  to  the  body  magnetized.  On  the  con- 
trary, it  is  a  molecular  condition  developed  within  the  body 
itself.  The  probable  nature  of  this  condition  is  suggested 
by  the  following  experiments. 

When  a  magnet  is  broken,  unlike  poles  are  produced 
at  the  broken  ends,  and  each  piece  becomes  a  complete 
magnet  (Fig.  350).  (A  magnetized  sewing  or  knitting 


FIG.  350.  —  Poles  of  a  Broken  Magnet. 

needle  is  convenient  for  the  experiment.)  A  test  with 
iron  filings  will  show  that  the  new  poles  are  as  strong  as 
the  original  ones.  The  magnet  may  be  broken  into  smaller 
and  smaller  pieces  indefinitely,  and  each  piece  will  still 
have  a  north  and  a  south  pole.  Since  the  act  of  breaking 
is  not  a  magnetizing  process,  it  follows  that  a  magnet  is 
magnetized  throughout  its  entire  length.  The  absence  of 
attracting  power  at  any  point,  as  at  the  center,  may  be 
regarded  as  due  to  the  equal  and  opposite  action  of  a  north 
and  a  south  pole  at  that  place.  When  these  poles  are  sep- 
arated by  breaking,  they  no  longer  neutralize  each  other.  A 
magnet  may  there- 
fore be  regarded  as 
composed  of  a  multi- 
tude of  little  mag-  &  ~  sw_.  s 

nets,  with   their   like    FlG-  35I-  — A  Magnet  is  Virtually  Composed  of  a 

Multitude  of  Smaller  Magnets. 

poles  pointing  in,  the 

same  direction  (Fig.  351).     If  the  intensity  of  magnetiza- 
tion were  the  same  at  all  points,  the  adjacent  unlike  poles 


SN' 


n         s 

n        s 

n       s 

n      s\n       s 

n      s 

n      s 

n      s 

n        s 

n       s 

n       s 

n      s\n       s 

n      s 

n      s 

L?l        S 

n        s 

n        s 

n       s 

n      s\n        s 

n        s 

n      s 

n      s 

464  MAGNETISM 

of  these  little  magnets  would  all  neutralize  one  another 
except  at  the  very  ends.  In  reality,  however,  the  mag- 
netization grows  weaker  toward  the  ends,  and  consequently 
the  poles  extend  some  distance  .back  from  them. 

Undoubtedly  the  smallest  visible  fragment  of  a  magnet 
is  itself  a  complete  magnet,  having  a  north  and  a  south 
pole;  and  this  is  probably  true  of  the  individual  molecules, 
for  any  action  that  is  known  to  affect  the  molecules  of  a 
body  also  affects  the  magnetism  of  a  magnet.  Thus  the 
strength  of  a  magnet  is  diminished  by  heating  it,  until,  at 
a  bright  red  heat,  it  is  completely  demagnetized.  A  mag- 
net is  also  weakened  by  any  mechanical  disturbance  of 
its  molecular  arrangement,  as  in  striking,  bending,  or  twist- 
ing it.  The  effect  of  bending  and  twisting  is  easily  shown 
with  a  magnetized  piece  of  iron  wire.  On  the  other  hand, 
a  piece  of  steel  becomes  more  strongly  magnetized  if  it  is 
hammered,  or  heated  and  allowed  to  cool,  while  it  is  near 
a  magnet. 

These  facts  and  others  of  a  similar  character  have  led 
to  the  theory  that  each  molecule  of  a  magnetic  substance 

is  a  permanent  magnet. 
In  an  unmagnetized 
body  these  molecular 


magnets    point    indis- 


FIG.  352.  -  Arrangement  of  Molecules  in  Criminately  in  all 

Unmagnetized  Iron  or  Steel.  tionS   (Fig.  352),   SO 

that  they  neutralize  each  other's  external  magnetic  effects. 
In  a  magnetized  body  the  greater  number  of  the  mole- 
cules lie  with  their  like  poles  pointing  in  the  same  general 
direction  (Fig.  353).  In  the  act  of  magnetizing  a  body 
the  molecules  are  turned  around,  more  or  less  completely, 
into  one  particular  direction.  If  all  the  molecules  were 
turned  in  the  same  direction,  the  limit  of  possible  magneti- 


PROPERTIES   OF   MAGNETS  465 

zation  would  be  reached.  Soft  iron  is  more  readily  mag- 
netized than  steel  because  its  molecules  are  more  easily 
turned  about,  and  it  loses  its  magnetism  more  readily  for 
the  same  reason. 

According  to  this  theory,  heat  weakens  a  magnet  because 
it  increases  molecular 
motion,  and  the  mole- 
cules jostle  one  another 
out  of  position.    Ham- 
mering,   bending,    and          FIG.  353- -  Arrangement  of  Molecules  in 
twisting     also     disturb  Magnetized  Iron  or  Steel. 

the  molecular  arrangement.  On  the  other  hand,  any  dis- 
turbance of  the  molecules  in  the  presence  of  a  magnetizing 
force  helps  to  turn  them  round  into  line  with  that  force. 
These  effects  may  be  illustrated  by  an  experiment  with 
steel  filings,  in  which  each  particle  represents  a  molecule 
on  a  greatly  magnified  scale.  A  test  tube  loosely  filled 
with  the  filings  is  held  in  a  horizontal  position,  while  the 
filings  are  .jarred  toward  one  end  by  repeatedly  tapping 
that  end  with  a  pole  of  a  magnet.  Testing  with  a  magnetic 
needle  will  show  that  the  mass  of  filings  now  has  a  pole  at 
each  end,  like  a  bar  magnet.  This  is  due  to  the  regular 
arrangement  of  the  magnetized  particles.  Shaking  the 
tube  destroys  this  arrangement,  and  the  mass  as  a  whole 
"  loses  its  magnetism,"  although  each  individual  particle 
is  still  a  magnet. 

PROBLEMS 

1.  (a)  When  a  pole  of  a  strong  magnet  is  brought  toward  the  like  pole 
of  a  magnetic  needle,  repulsion  may  be  followed  by  attraction  as  the  mag- 
net is  brought  closer.     Explain.     (6)  The  same  may  happen  when  an  end 
of  a  weakly  magnetized  piece  of  iron  is  brought  toward  a  needle.     Explain. 

2.  Why  should  decision  as  to  the  polarity  of  a  magnetized  body  be  based 
on  repulsion  of  the  magnetic  needle  rather  than  on  attraction? 

3.  In  what  different  ways  may  an  unmagnetized  magnetic  substance 
be  distinguished  from  a  magnet? 


466  MAGNETISM 

II.    THE  MAGNETIC  FIELD 

375.   Magnetic    Lines    of    Force.  —  When   a   magnetic 
needle  is  near  a  magnet,  as  at  O  (Fig.  354),  its  north  pole 

is  attracted  by  the  south 
pole  of  the  magnet  and  re- 
pelled by  the  north  pole. 
These  forces  are  represented 
in  magnitude  and  direction 
by  OB  and  OA  respectively, 

FIG.  354.  —  Magnetic  Line  of  Force.  .     .  .   . 

O  being  the  position  of  the 

north  pole  of  the  needle.  The  attraction  is  the  greater 
force,  since  it  is  due  to  the  nearer  pole.  The  resultant  of 
these  two  forces  is  represented  by  OR  (found  by  constructing 
the  parallelogram  of  forces).  Hence  the  magnetic  needle 
behaves  as  if  its  north  pole  were  acted  upon  by  the  single 
force  OR.  If  the  south  pole  of  the  needle  is  at  0,  the  result- 
ant force  upon  it  is  equal  and  opposite  to  OR.  Obviously 
the  two  poles  of  the  needle  can  not  be  at  0  at  the  same  time ; 
but  if  the  needle  is  very  short  and  its  center  is  at  0,  the 
forces  acting  on  its  poles  are  approximately  as  stated,  and 
the  needle  will  come  to  rest  with  its  north  pole  pointing 
in  the  direction  OR,  this  being  the  position  of  equilibrium. 
If  the  needle  is  moved  constantly  in  the  direction  in 
which  its  north  pole  points,  it  will  trace  the  curved  path 
OCS.  Starting  from  N,  the  entire  curve  NOCS  can  be 
traced  in  this  manner.  This  curve  is  called  a  magnetic 
line  of  force.  Going  from  N  toward  S,  its  direction  at 
every  point  is  the  direction  of  the  resultant  magnetic  force 
at  that  point  upon  the  north  pole  of  the  needle.  Going 
from  S  toward  N,  its  direction  at  every  point  is  the  direc- 
tion of  the  resultant  magnetic  force  at  that  point  upon  the 
south  pole  of  the  needle. 


THE  MAGNETIC  FIELD  467 

Lines  of  force  are  of  great  importance,  and  we  shall 
meet  with  them  frequently  in  the  study  of  electricity.  To 
save  words,  it  is  always  understood  that  the  expression 
the  direction  of  a  line  of  force  means  the  direction  of  the 
magnetic  force  upon  the  north  pole  of  a  magnetic  needle 
at  any  point  along  the  line.  Thus,  in  the  present  in- 
stance, the  direction  of  the  line  of  force  is  from  N  toward 
S.  Any  number  of  lines  of  force  can  be  traced  about  a 
magnet,  in  the  manner  above  described.  In  general,  the 
direction  of  the  line  of  force  passing  through  any  point 
within  the  range  of  action  of  a  magnet  is  the  direction  in 
which  the  north  pole  of  a  short  magnetic  needle  points 
at  that  place. 

To  explore  the  entire  space  about  a  magnet  by  this  method  is  a 
long  and  tedious  process;  but  the  same  information  can  be  obtained 
in  a  very  simple  and  striking  manner  by  means  of  iron  filings.  The 
filings  are  sifted  upon  a  sheet  of  cardboard,  laid  over  the  magnet. 
Each  particle  of  iron  becomes  magnetized  and  tends  to  place  itself 
lengthwise  along  a  line  of  force.  Tapping  the  cardboard  with  the 
finger  assists  the  magnetic  forces  by  overcoming  friction.  The  fil- 
ings cling  together  in  somewhat  irregular,  broken  lines,  which  never- 
theless ind;cate  the  lines  of  force  very  clearly  (Fig.  355^).  The  lines 
of  force  are  really  smooth,  unbroken  curves,  and  are  continuous  with 
lines  of  magnetic  induction  within  the  magnet  (Fig.  3556).  Each  line 
of  force  and  the  corresponding  line  of  induction  together  form  a  closed 
curve.  (Lines  of  force  are  often  represented  by  dotted  lines,  as  in 
some  of  the  diagrams  that  follow.) 

376.  The  Magnetic  Field.  —  Any  space  within  which 
magnetic  forces  act,  when  magnetic  material  is  present  to 
be  acted  upon,  is  called  a  magnetic  field.  Every  magnet 
is  surrounded  by  a  magnetic  field,  which  extends  indefi- 
nitely in  all  directions.  Practically,  it  is  regarded  as  extend- 
ing only  as  far  as  the  magnet  noticeably  affects  a  magnetic 
needle.  The  region  within  which  the  field  is  strong  enough 
to  turn  iron  filings  into  line  is  smaller  than  this. 


468 


MAGNETISM 


There  are  other  magnetic  fields  than  those  of  magnets. 
We  shall  find  later  on  that  a  current  of  electricity  always 
produces  a  magnetic  field  about  the  wire  or  other  conduc- 


FIG.  3550.  —  Lines  of  Force  in  the  Field  of  a  Bar  Magnet. 

tor  in  which  the  current  is  flowing.  Everywhere  upon  the 
earth's  surface  a  magnetic  needle  sets  itself  in  a  definite 
direction,  when  no  magnet  or  electric  current  is  near. 
This  behavior  shows  that  the  earth  is  surrounded  by  a 
magnetic  field,  as  if  it  were  a  huge  magnet.  This  field  is 
much  too  weak  to  direct  iron  filings,  and  so  does  not 


FIG. 


.  —  Lines  of  Force  and  Lines  of  Magnetic 
Induction. 


interfere  with  their  use  in  studying  the  fields  of  magnets. 

A  magnetic  field  is  to  be  regarded  as  having  an  actual 

physical  existence.     The  portion  of  space  that  it  occupies 

possesses  properties  which  other  space  does  not.     These 


THE  MAGNETIC  FIELD 


469 


AA/fAAAAAAAAAA'AAA/f 
il'  IAI    I 

j 


A  B 

FIG.    356.  —  Magnetic    Ac- 
tion in  a  Uniform  Field. 


properties  are  often  considered  without  reference  to  the  ori- 
gin of  the  field.  Thus  we  say  that  a  magnetic  needle,  when 
placed  in  a  magnetic  field,  tends  to  set  itself  parallel  to  the 
lines  of  force  of  the  field. 

The  properties  of  a  magnetic  field 
with  which  we  are  principally  con- 
cerned are  its  intensity,  or  strength, 
and  the  direction  of  its  lines  of  force. 
A  diagram  or  map  of  a  field  indi- 
cates the  relative  intensities  of  its 
different  parts  by  the  relative  distances  between  the  lines 
of  force.  Where  the  lines  run  closer  together  the  field  is 
stronger;  where  they  are  more  widely  separated  it  is  weaker 
(Fig.  3556).  In  a  field  of  uniform  intensity  the  lines  are 
straight  and  parallel,  and  are  equally  spaced.  Any  limited 
portion  of  the  earth's  field  is  a  good  example  (Fig.  356).  In 

such  a  field  the  forces  act- 
ing upon  the  poles  of  a 
magnetic  needle  are  ex- 
actly equal  and  opposite. 
Together  they  form  a 
couple,  which  causes  rota- 
tion when  the  needle  is  at 
an  angle  with  their  lines  of 
action;  but  they  do  not 
tend  to  move  the  needle  as 
a  whole  in  either  direction. 

377.  Other  Properties 
of    Magnetic    Fields.  - 

When  the  north  pole  of 
one  magnet  is  placed 
near  the  south  pole  of 


FIG.    357.  —  Magnetic     Attraction     along 
Lines  of  Force. 


470 


MAGNETISM 


another,  many  of  the  lines  of  force  extend  across  be- 
tween them  (Fig.  357);  when  like  poles  are  adjacent  the 
lines  in  one  field  turn  away  from  those  in  the  other  (Fig. 
358).  These  are  typical  cases.  In  general,  we  find  lines 

of  force  extending  from  the 
north  pole  of  a  magnet  to 
the  south  pole  of  the  same 
or  to  the  south  pole  of  an- 
other magnet;  but  there  are 
no  lines  connecting  like 
poles.  Neither  do  lines  of 
force  ever  cross  each  other; 
for  at  any  point  of  inter- 
section the  magnetic  forces 
would  have  two  resultants, 
which  is  not  true  of  any 
set  of  forces. 

Any  magnetic  body  placed 
in  a  magnetic  field  modifies 
the  field  and  alters  the  dis- 
tribution of  the  lines  of 
force.  The  effect  of  a  soft 
iron  bar  in  a  uniform  field  is  shown  in  Figure  359.  The 
lines  of  force  crowd  together,  entering  the  iron  at  one  end 
and  leaving  it  at  the  other.  It  is  as  if  the  iron  afforded 
an  easier  path  for  the  lines  than  air  does;  i.e.  the  iron 
has  greater  magnetic  permeability  than  air  (Art.  373). 
Another  way  of  stating  it  is  that  the  iron  becomes  mag- 
netized by  induction  (with  the  polarity  shown  in  the 
figure),  and  adds  its  own  field  to  the  original  one. 

378.  Theory  of  Magnetic  Action.  —  When  an  object  is  pushed 
with  a  stick  or  pulled  with  a  rope,  the  mechanism  of  the  action  is 
clear.  The  stick  or  the  rope  serves  to  transmit  the  force  from  the 


m 


FIG.  358.  —  Magnetic  Repulsion  across 
Lines  of  Force. 


THE    MAGNETIC    FIELD  471 

hand  to  the  body  acted  upon.  The  stick  sustains  a  compressive 
stress,  which  tends  to  bend  it;  and  the  rope  a  tensile  stress,  which 
tends  to  pull  it  apart.  Magnetic  forces  act  between  bodies  at  a  dis- 
tance, without  the  visible  aid  of  any  intervening  medium,  and  this  action 
takes  place  in  a  vacuum  as  readily  as  in  air.  In  these  respects  mag- 
netic action  is  like  gravitation;  and  both  are  inversely  proportional 
to  the  square  of  the  distance.  In  other  respects  they  are  very  dif- 
ferent. The  force  of  gravitation  is  always  an  attraction,  and  it  acts 
on  all  masses  irrespective  of  their  material.  Magnetic  action  is  lim- 
ited to  magnetic  materials,  and  the  force  may  be  either  an  attrac- 
tion or  a  repulsion.  Magnetic  action  can  be  cut  off  by  a  magnetic 
screen.  Gravitation  is  unaffected  by  any  intervening  medium. 
Gravitational  attraction  is  excessively  small  between  masses  of  ordi- 
nary size.  Magnetic  forces  are  relatively  enormous. 

No  satisfactory  theory  of  gravitation  has  yet  been  proposed;  but 
it  is  very  probable  that  the  attraction  takes  place  through  the  medium 


FIG.  359.  —  Effect  of  Soft  Iron  in  a  Magnetic  Field. 

of  the  ether.  That  the  ether  is  the  medium  through  which  magnetic 
forces  act  is  hardly  a  matter  of  doubt.  The  action  appears  to  be 
in  the  nature  of  a  tension  along  .the  lines  of  force  and  a  pressure  at 
right  angles  to  them,  as  if  the  ether  were  an  elastic  solid  (Art.  296) 
in  a  state  of  strain.  The  ether  in  this  condition  may  be  compared 
to  a  stretched  piece  of  rubber,  which  tends  to  shorten  and  to  become 
thicker.  The  tension  along  the  lines  of  force  draws  unlike  magnetic 
poles  together  (Fig.  357),  and  the  pressure  at  right  angles  to  the  lines 
pushes  like  poles  apart  (Fig.  358). 


472  MAGNETISM 

III.  THE  EARTH'S  MAGNETIC  FIELD 

379.  Magnetic  Meridians  and  Declination.  —  Every- 
where upon  the  earth's  surface  a  magnetic  needle,  when 
removed  from  all  magnetic  bodies,  comes  to  rest  in  a  defi- 
nite direction,  clearly  indicating  that  it  is  controlled  by  a 
magnetic  field.  This  is  the  magnetic  field  of  the  earth. 
Its  cause  is  not  very  well  understood;  but  it  is,  probably 
due  to  electric  currents  circulating  round  the  earth.  Large 
masses  of  iron  ore  produce  local  variations  in  the  field,  but 
they  are  evidently  not  its  primary  cause. 


FIG.  360.  —  Magnetic  Meridians. 

A  line  extending  over  the  earth's  surface  and  having  at 
every  point  the  direction  of  the  magnetic  needle  is  called 
a  magnetic  north-and-south  line,  or  a  magnetic  meridian. 
Magnetic  meridians  are  represented  in  Fig.  360  by  the  heav- 
ier lines.  They  are  more  or  less  irregular,  and  are  nearly 
everywhere  at  a  considerable  angle  with  the  geographical 
meridians,  or  the  true  north-and-south  lines.  This  angle 
is  called  the  magnetic  decimation,  or,  simply,  the  declina- 
tion. In  the  eastern  part  of  America  the  declination  is 
toward  the  west;  in  the  western  part  it  is  toward  the  east. 


THE    EARTH'S    MAGNETIC    FIELD 


473 


A  line  connecting  all  points  where  the  declination  is  the 
same  is  called  an  isogonic  line  (Greek  isos,  equal,  and  gonia, 
angle).  Such  lines  are  irregular  curves  (Fig.  361). 

Magnetic  declination  is  subject  to  daily  and  annual 
variations,  amounting,  however,  only  to  a  small  fraction 
of  a  degree.  There  is  also  a  slow  but  continuous  change  in 
one  direction  from  year  to  year.  At  London,  England, 
the  declination  in  1580  was  11°  east;  in  1800  it  was  24° 
west.  Since  the  latter  date  the  change  has  been  in  the 
opposite  direction. 


150  120  80  60  80  0 


FIG.  361.  —  Chart  of  Isogonic  Lines. 

380.  Magnetic  Inclination  or  Dip.  Magnetic  Poles  of  the  Earth. 
—  The  ordinary  magnetic  needle  is  free  to  turn  only  in  a  horizontal 
plane.  Since  its  center  of  gravity  is  below  the  point  of  support,  its 
weight  opposes  any  downward  tilting  of  either  end.  Hence  we  can 
not  tell  from  the  behavior  of  such  a  needle  whether  the  lines  of  force 
of  the  earth's  field  are  horizontal  or  inclined.  For  this  purpose  we 
require  a  dipping  needle,  which  is  a  magnetic  needle  mounted  on  a 
horizontal  axis  through  its  center  of  gravity,  and  sometimes  also 


474  MAGNETISM 

suspended  from  an  untwisted  fiber,  which  serves  as  a  vertical  axis 
(Fig.  362).  A  needle  thus  mounted  is  free  to  assume  any  direction, 
and  its  direction  is  wholly  unaffected  by  gravity.  It  will,  therefore, 
come  to  rest  parallel  to  the  lines  of  force  of  the  earth's  field. 

The  angle  between  the  direction  of  the  dipping  needle  and  the  hori- 
zontal is  called  the  magnetic  inclination  or  dip.  The  irregular  lines 
extending  across  Fig.  363  are  lines  of  equal  dip.  The  line  of  no  dip 
is  called  the  magnetic  equator.  North  of  the  magnetic  equator  the 
north  pole  of  the  needle  is  depressed,  and  south  of  it  the  south  pole. 
Arctic  explorers  have  found  a  place  where  the  dip  is  90°.  This 
is  the  north  magnetic  pole  of  the  earth  (so  called  from  its  geographical 
position;  its  polarity  is  like  that  of  the  south  pole  of  a 
magnet).  It  is  nearly  1400  mi.  from  the  geographical 
north  pole,  and  is  shown  in  Fig.  360  as  the  point  in 
the  northern  hemisphere  to  which  the  magnetic  me- 
ridians converge.  It  is  situated  in  latitude  70°  5'  N. 
and  longitude  96°  43'  W.  The  south  magnetic  pole 
was  discovered  in  1908  by  an  expedition  under  the 

FlG      62 command   of  Lieutenant  Shackleton  of  the  British 

The    Dip-     navy.    It  lies  in  latitude  72°  25'  S.  and  longitude  154° 
ping  Needle.     E   strictly  speaking,  the  magnetic  poles  of  the  earth 
lie  far  below  the  surface. 

381.  Intensity  of  the  Earth's  Magnetic  Field.  Induct- 
ive Action.  —  The  magnetic  field  of  the  earth  is  relatively 
very  weak,  that  of  an  ordinary  magnet  being  thousands  of 
times  stronger;  but  its  inductive  action  is  sufficient  to  pro- 
duce considerable  magnetization  in  iron  and  steel.  This 
is  readily  shown  with  a  long  rod  of  soft  iron  (Norway  iron). 
While  the  rod  is  held  in  a  north-and-south  line,  or,  better, 
at  the  angle  of  dip,  it  will  be  found  to  be  magnetized,  with 
a  north  pole  at  its  north  or  lower  end.  On  reversing  the 
rod,  its  polarity  is  also  instantly  reversed,  provided  the 
iron  is  very  soft;  otherwise  it  may  be  necessary  to  strike 
the  rod  on  the  end  while  it  is  held  in  position. 

Any  mass  of  iron  or  steel  that  remains  in  one  position  for 
a  time  becomes  magnetized  by  the  earth's  inductive  action, 


THE  EARTH'S  MAGNETIC  FIELD 


475 


especially  if  it  is  subjected  to  jarring,  as  in  railroad  tracks 
and  bridges.  The  magnetism  of  lodestones  has  doubtless 
been  produced  by  the  same  cause. 

382.  Importance  of  the  Earth's  Magnetic  Field.  Mag- 
netic Surveys.  —  The  earth's  magnetic  field  is  of  the  great- 
est importance,  since  the  use  of  the  compass  in  determining 
directions  on  land  and  sea  depends  upon  it.  A  compass 
is  a  magnetic  needle  suitably  mounted  within  a  box,  to- 
gether with  a  compass  card  or  dial.  In  the  mariner's  com- 


FIG.  363.  —  Lines  of  Equal  Dip. 

pass  the  card  turns  with  the  needle,  so  that  at  all  times  it 
correctly  indicates  the  directions  marked  upon  it  (Fig. 
364),  allowance  being  made  for  the  declination.  In  the 
surveyor's  compass  the  dial  is  in  the  bottom  of  the  box 
and  the  needle  moves  over  it. 

The  true  north  can  be  determined  with  a  compass  only  when  the 
declination  at  the  place  is  known.  This  is  given  by  a  declination  map 
or  chart  of  the  region,  which  should  be  as  accurate  as  possible.  Owing 


476 


MAGNETISM 


to  the  continuous  change  in  the  earth's  magnetism,  new  magnetic 
charts  must  be  constructed  from  time  to  time;  and  in  order  to  obtain 
the  necessary  information  for  this  purpose,  the  civilized  nations  of 
the  world  are  constantly  making  magnetic  surveys  on  land  and  sea. 
In  the  United  States  this  work  is  done  by  the  Division  of  Terrestrial 

Magnetism  of  the  United  States 
Coast  and  Geodetic  Survey.  In 
a  report  recently  published  the 
Survey  gives  maps  and  tables 
constructed  from  observations 
made  at  over  3300  stations  over 
two  thirds  of  which  were  occu- 
pied by  the  Survey  from  1899 
to  1906.  The  Carnegie  Insti- 
tution at  Washington,  D.  C., 
has  undertaken  a  series  of  sur- 
veys to  determine  the  magnetic 
conditions  over  all  the  oceans. 
For  the  greatest  accuracy  the 
vessel  in  which  such  work  is 
carried  on  must  be  as  nearly 
non-magnetic  as  possible;  for  all 

iron  and  steel  parts  of  a  ship  become  magnetized  by  the  induction 
due  to  the  earth's  field,  and  this  magnetism  affects  the  compass 
needle  more  or  less.  A  ship  has  been  built  especially  for  this  ser- 
vice (1909).  It  is  constructed  entirely  of  non-magnetic  materials, 
with  the  exception  of  certain  parts  of  the  engine.  The  fasten- 
ings consist  of  locust-wood  nails,  copper  and  bronze  bolts,  and 
composition  spikes.  All  metal  deck  fittings  and  metal  work  on 
spars  and  rigging  are  of  bronze,  copper,  or  gun-metal. 


FIG.  364.  —  The  Compass  Card.  Recit- 
ing the  names  of  the  thirty-two 
points  is  called  by  sailors  "Boxing 
the  Compass." 


CHAPTER  XII 
ELECTROSTATICS 

383.  Introduction.  —  This  is  often  called  the  electrical 
age,  and  with  good  reason;  for  electricity  is  now  doing  a 
large  part  of  the  work  of  the  world,  and  that  part  is  increas- 
ing rapidly  from  year  to  year.  The  use  of  electrical  energy 
in  transportation,  in  driving  the  machinery  of  shops  and 
factories,  in  lighting  buildings  and  city  streets,  in  trans- 
mitting messages  by  telegraph  and  telephone,  etc.,  is  more 
or  less  familiar  to  every  one.  All  this  has  been  accomplished 
within  the  past  century,  through  the  discovery  and  appli- 
cation of  the  laws  of  electrical  action,  but  without  a 
knowledge  of  what  electricity  really  is.  The  earlier  ideas 
concerning  the  nature  of  this  wonderful  agent  have  been 
discarded.  In  recent  years  rapid  progress  has  been  made 
in  the  development  of  a  new  theory,  which  is  supported 
by  such  an  array  of  facts  that  it  promises  to  be  final.  Of 
this  we  shall  have  something  to  say  in  the  concluding 
chapter.  Meanwhile  we  shall  be  mainly  concerned  with 
matters  of  fact  —  with  electrical  phenomena,  their  laws, 
and  their  applications. 

The  subject  of  electricity  is  divided  into  two  parts, 
electrostatics  dealing  with  electricity  at  rest,  or  in  equi- 
librium, and  electrodynamics,  dealing  with  electricity 
in  motion.  We  shall  begin  with  electrostatics,  which 
is  the  older  branch  of  the  science,  the  period  of  its 
greatest  development  being  the  eighteenth  century. 

477 


478  ELECTROSTATICS 

384.  Electrification    by   Friction.  —  When    a   vulcanite 
(hard  rubber)  rod  is  rubbed  with  fur  or  flannel,  it  acquires 
the  power  of  attracting  bits  of  paper  or  pith,  and  other  light 
bodies.     The  same  results  are  obtained  in  greater  or  less 
degree  with  many  different  substances,  e.g.  with  sealing 
wax,  resin,  or  sulphur  when  rubbed  with  fur  or  flannel, 
and  with  a  glass  rod  when  rubbed  with  silk.     In  all  cases 
the  bodies  must  be  dry,  and  the  drier  the  atmosphere  the 
better. 

It  was  known  to  the  ancient  Greeks  that  amber  possesses 
this  power  of  attraction  when  rubbed;  but  Dr.  Gilbert, 
an  English  physician  and  scientist  of  the  sixteenth  century, 
seems  to  have  been  the  first  to  make  a  systematic  study 
of  the  phenomenon.  In  his  great  work  on  magnetism, 
published  in  1600,  he  called  all  substances  which  he  had 
found  to  exhibit  this  property  of  amber  "electrics,"  after 
elektron,  the  Greek  name  for  amber.  He  described  the 
condition  of  the  rubbed  body  as  a  state  of  electrification,  and 
called  the  force  exerted  by  it  electric  attraction.  The  agent 
to  which  electrostatic  phenomena  are  due  became  known 
as  electricity.  A  body  in  a  condition  to  exert  electric  at- 
traction is  said  to  be  electrified  or  to  have  an  electric 
charge,  or  to  be  charged. 

385.  Positive  and   Negative  Electrification.  —  An  elec- 

trified  vulcanite  rod  suspended  by  a 
thread  turns  away  when  another  elec- 
trified vulcanite  rod  is  brought  near  it 
showing  that  it  is  repelled  (Fig.  365). 
Two  electrified  glass  rods  also  exhibit 
FIG.  365.  —  Electrostatic"  repulsion   when   tested    in    the    same 
manner;  but  a  vulcanite  rod  rubbed 
with  fur  and  a  glass  rod  rubbed  with  silk  attract  each  other. 


ELECTROSTATICS  479 

Any  electrified  body  either  attracts  the  electrified  glass 
rod  and  repels  the  electrified  vulcanite,  or  it  repels  the 
glass  and  attracts  the  vulcanite.  From  this  we  learn 
that  there  are  two,  and  only  two,  states  of  electrification 
or  two  kinds  of  electric  charges.  The  electrification  of  glass 
when  rubbed  with  silk,  is  called  positive,  and  the  glass  is 
said  to  have  a  positive  charge,  or  to  be  positively  eletri- 
fied.  A  negative  charge  is  one  like  that  of  vulcanite  when 
rubbed  with  fur  or  flannel. 

The  law  of  electrostatic  action,  as  shown  by  the  experi- 
ments described  above,  is  that  like  charges  repel  and  unlike 
charges  attract  each  other.  The  force,  whether  of  attrac- 
tion or  repulsion,  becomes  less  as  the  distance  between  the 
charges  is  increased. 

386.   Conductors  and  Non-conductors  or  Insulators.  — 

Gilbert  was  unable  to  electrify  the  metals  and  some  other 
substances,  and  he  therefore  called  them  "non-electrics." 
It  was  later  discovered  that  such  bodies  are  conductors  of 
electricity,  and  permit  the  charge  to  escape  through  the 
body  of  the  experimenter  to  the  earth  as  fast  as  it  is  formed. 
Vulcanite  and  other  substances  which  retain  their  charges 
are  called  non-conductors,  or  insulators.  A  metal  rod  or 
other  conducting  substance  can  be  electrified  if  the  precau- 
tion is  taken  to  interpose  some  insulating  body  between  it 
arid  the  hand. 

All  substances  may  be  roughly  classified  as  conductors 
or  non-conductors  of  electricity;  but  there  is  no  sharp  divid- 
ing line  between  them.  The  two  classes  merge  impercep- 
tibly into  each  other  when  the  substances  are  arranged  in 
the  order  of  their  electrical  conductivity,  just  as  in  the  case  of 
good  and  poor  conductors  of  heat.  At  the  one  extreme  we 
have  the  best  conductors,  among  which  are  the  metals  and 


480  ELECTROSTATICS 

solutions  of  salts  and  acids  in  water;  and,  at  the  other  ex- 
treme, the  best  insulators,  such  as  vulcanite,  rubber,  sul- 
phur, shellac,  glass,  paraffin,  sealing  wax,  silk,  and  air. 
Wood,  cotton,  and  various  other  substances  occupy  an 
intermediate  position. 

If  the  charged  bodies  in  electrostatic  experiments  are 
not  themselves  good  insulators,  they  must  have  a  non- 
conducting support,  to  prevent  the  escape  of  the  charge 
to  neighboring  bodies.  A  body  thus  supported  is  said 
to  be  insulated.  It  should  be  noted  that  a  given  material 
may  be  a  sufficiently  good  insulator  under  certain  con- 
ditions but  not  under  others.  This  is  familiar  in  the  vari- 
ous uses  of  the  electric  current.  A  cotton  covering  suffices 
for  the  wire  used  in  the  circuits  of  electric  bells,  telegraph 
instruments,  and  the  like.  Better  insulation  is  afforded 
by  a  thick  covering  of  cotton  and  rubber,  as  in  electric- 
light  circuits.  In  spark  coils  and  similar  appliances, 
where  the  conditions  are  very  exacting,  the  wire  is  covered 
with  silk.  In  controlling  electric  charges  the  best  insu- 
lators are  required,  such  materials  as  cotton  and  wood  being 
wholly  inadequate. 

387.  Charging  and  Discharging  by  Conduction.  —  TJie 
bits  of  paper  or  other  light  bodies  that  cling  to  an  electri- 
fied rod  often  dart  away  after  brief  contact.  While  the 
paper  is  in  contact  with  the  rod,  it  receives  a  portion  of 
the  charge  by  conduction,  and  is  then  driven  off  by  the 
repulsion  between  its  charge  and  the  like  charge  of  the  rod. 
Since  the  rod  is  a  non-conductor,  it  parts  with  its  charge 
very  slowly,  even  at  points  of  actual  contact.  On  this 
account  the  papers  that  chance  to  touch  at  only  a  few 
points  are  not  repelled. 

This  action  is  shown  to  better  advantage  with  a  pith 


ELECTROSTATICS  48 1 

ball,  suspended  by  a  silk  thread.     The  ball  swings  out 
toward  an  electrified  rod,  and  rolls  about  over  it,  taking 
up  the  charge,  until  it  is  repelled  (Fig.  366).     The  repul- 
pulsion  continues,  since  the 
charge  on  the  ball  can  not 
escape  through  the  thread; 
but,  if  the  ball  is  permitted 
to  come   in   contact  with 
the   metal  or  wood   sup- 
port from  which  it  hangs, 
or  is  touched  with  the  fin-       FlG  366  _  pith  Ball    a  Attracted 

ger,  its  charge  is  Conducted          before   Touching  Electrified  Rod;  b, 
...  .  Repelled  after  Touching. 

away  and  it  is  again   at- 
tracted by  the  rod.     If  the  ball  is  suspended  by  a  cotton 
thread,  it  is  not  repelled  at  all,  for  the  cotton  conducts 
the  charge  away  as  fast  as  the  ball  receives  it. 

An  electrified  non-conductor  is  discharged  by  bringing 
every  part  of  its  surface  in  contact  with  a  conductor. 
Wiping  the  surface  with  the  hand  is  sufficient. 

388.  Electroscopes.  —  An  instrument  which  shows 
whether  a  body  has  an  electric  charge,  and,  if  so,  whether 
the  charge  is  positive  or  negative,  is  called  an  electroscope. 
The  simplest  form  of  electroscope  is  a  pith  ball  suspended 
by  a  silk  thread.  The  ball  is  first  given  a  charge  of  known 
kind,  either  positive  or  negative,  as  may  be  desired,  and 
the  body  to  be  tested  is  then  brought  near  it.  If  the  ball  is 
repelled,  the  body  is  electrified,  and  its  charge  is  like  that 
of  the  ball.  Attraction  is  not  a  reliable  test;  for  an  un- 
charged body  will  attract  the  ball,  as  may  be  shown  by 
bringing  a  finger  near  it. 

The  gold-leaf  electroscope  is  a  much  more  sensitive 
instrument  (Fig.  367).  Its  conducting  parts  consist  of  a 


482 


ELECTROSTATICS 


metal  rod,  with  a  knob  or  a  disk  at  the  top  and  two  leaves 
of  gold  foil  at  the  lower  end.     The  leaves  are  inclosed  in 

a  box  or  a  flask,  to  protect  them 
from  currents  of  air  and  from 
mechanical  injury.  The  stopper 
through  which  the  rod  passes  is 
of  some  non-conducting  mate- 
rial, and  serves  as  an  insulating 
support.  When  the  leaves  are 
charged  they  spread  apart,  owing 
to  their  mutual  repulsion,  for 

FIG.  367. -Electroscope.  ^     charges     ^      necessarily 

both  positive  or  both  negative.     The  use  of  this  instru- 
ment depends  upon  electrostatic  induction. 

389.  Electrostatic  Induction.  —  As  a  charged  rod  is 
brought  toward  the  knob  of  an  electroscope,  the  leaves 
diverge  more  and  more;  when  the  rod  is  removed,  they 
drop  together  again.  The  presence  of  the  charged  rod 
(without  contact)  produces  an  unlike  charge  on  the  knob 
of  the  electroscope  and  an  equal  like 
charge  on  the  leaves  (Fig.  368).  This 
action  is  called  electrostatic  induction, 
and  the  resulting  charges  are  called 
induced  charges.  When  the  rod  is 
removed  these  charges  disappear,  and 
the  leaves  no  longer  repel  each  other. 

Induction  and  electrostatic  phe- 
nomena in  general  can  be  explained 
if  we  adopt  the  theory  that  an  unelectrified  body  possesses 
equal  quantities  of  positive  and  negative  electricity, 
which,  being  equally  distributed  over  the  body,  exactly 
neutralize  each  other.  When  a  negatively  charged  rod  is 


FIG.  368.  —  Induction  in 
Knob  and  Leaves. 


ELECTROSTATICS  483 

brought  near  an  electroscope,  as  in  the  above  experiment, 
it  attracts  the  positive  electricity  to  the  knob  and  repels 
the  negative  to  the  leaves.  As  soon  as  the  rod  is  removed 
the  induced  charges  are  brought  together  again  by  their 
mutual  attraction.  When  the  inducing  charge  is  positive 
the  negative  electricity  is  attracted  to  the  knob  and  the 
positive  repelled  to  the  leaves. 

The  attraction  of  unelectrified  \ 
bodies,    as    pith    balls,    bits    of  \ 
paper,   etc.,  is  due  to  induction      \  ,T^<^ 
(Fig.  369).     Since  the  unlike  in-          (i** 
duced  charge  is  the  nearer,  the 
attraction  is  greater  than  the  re- 
pulsion,  and  the  resultant  force  FIG.  369.  —  inductive  Action  on 

,.,         .  Pith  Ball. 

is  an  attraction.     Equal  and  op- 
posite induced  charges  always  appear  upon  an  insulated 
conductor  in  the  presence  of  a  charge  on  a  neighboring 
body. 

According  to  present  theory,  there  are  really  two  kinds  of  elec- 
tricity, but,  in  a  solid  conductor,  only  the  negative  electricity  is  free 
to  move.  If  this  view  is  correct,  a  body  becomes  positively  charged 
by  losing  some  of  its  negative  electricity,  and  negatively  charged  by 
receiving  an  excess  of  negative  electricity.  It  would  seem  more 
appropriate,  in  the  light  of  present  knowledge,  to  call  negative  elec- 
tricity positive  and  vice  versa,  since  the  negative  electricity  is  appar- 
ently the  freer  and  more  active  agent;  but,  as  the  terms  are  purely 
arbitrary  any  way,  it  does  not  matter. 

390.  Charging  by  Induction.  —  In  the  above  experiment 
with  the  electroscope  the  opposite  charges  induced  on  the 
knob  and  the  leaves  are  temporary,  in  the  sense  that  they 
disappear  as  soon  as  the  inducing  charge  is  removed.  The 
electroscope,  or  in  fact  any  insulated  conductor,  can  be 
permanently  charged  by  induction,  the  charge  being  per- 
manent in  the  sense  that  it  continues  after  the  inducing 


484  ELECTROSTATICS 

charge  is  removed.  To  charge  the  electroscope  by  induc- 
tion we  proceed  as  follows:  An  electrified  rod,  having,  let 
us  say,  a  negative  charge,  is  brought  near  the  knob. 
The  induced  charge  on  the  knob  is  positive,  that  on  the 
leaves  negative;  and  the  leaves  diverge  (Fig.  368).  While 
the  inducing  charge  is  still  present,  the  knob  is  touched 
with  the  finger.  The  leaves  instantly  fall  together,  show- 
ing that  their  negative  charge  is  lost.  It  has,  in  fact,  been 
conducted  away  through  the  finger  and  body  of  the  experi- 
menter to  the  earth.  Owing  to  the  repulsion  of  the  nega- 
tive inducing  charge,  the  negative  charge  of  the  leaves 
has  been  driven  away  as  far  as  possible.  The  positive 
charge  of  the  knob  can  not  escape,  although  a  conductor 
is  provided,  for  it  is  held  or  " bound"  by  the  attraction 
gf  the  negative  charge  on  the  rod.  The  finger  is  now  re- 
moved, and  afterward  the  rod.  As  the  rod  is  removed 
the  leaves  again  diverge.  They  are  now  positively  charged ; 
for  the  positive  charge  of  the  knob,  when  freed  from  the 
attraction  of  the  inducing  charge,  is  shared  with  the  leaves. 
The  electroscope  now  has  a  permanent  positive  charge. 

This  fact  can  be  tested  by  again  bringing  up  the  nega- 
tively charged  rod;  for,  as  the  rod  approaches,  the  leaves 
gradually  drop,  showing  that  their  positive  charge  has 
again  been  attracted  to  the  knob.  On  the  other  hand,  if 
a  positive  charge  is  brought  up,  the  divergence  of  the  leaves 
increases.  (Why?) 

391.  Use  of  the  Electroscope  in  Testing  Charges.  —  This 
behavior  of  a  charged  electroscope  serves  as  a  ready  means 
of  determining  the  kind  of  charge  on  any  body.  For  this 
purpose  the  charge  on  the  electroscope  must  be  of  known 
kind,  either  positive  or  negative.  If  the  charge  to  be  tested 
causes  greater  divergence  of  the  leaves,  as  it  is  brought 


ELECTROSTATICS  .    485 

near,  it  is  of  the  same  kind  as  the  charge  of  the  electro- 
scope; if  it  decreases  the  divergence  of  the  leaves,  it  is  of 
the  opposite  kind.  (Why?) 

392.  Positive  and  Negative  Electricities  always  Pro- 
duced in  Equal  Quantities.  —  As  already  stated  (Art.  389), 
the  inductive  action  of  a  neighboring  charge  always  pro- 
duces equal  positive  and  negative  charges  on  an  insulated 
conductor.  The  equality  of  the  charges  is  shown  by  the 
fact  that  they  exactly  neutralize  each  other  when  they 
reunite,  after  the  inducing  charge  is  removed.  If  the 
conductor  is  not  insulated,  the  two  electricities  are  still 
produced  in  equal  quantities,  but  the  repelled  charge  is 
conducted  away  and  lost,  e.  g.  when  the  electroscope  was 
touched  with  the  finger  in  the  presence  of  the  charged  rod. 

Friction  also  produces  both  kinds  of  electricity  and  in  equal  quan- 
tities. In  order  to  show  this,  precautions  must  be  taken  to  prevent 
the  escape  of  the  charge  from  either  of  the  two  bodies  which  are  rubbed 
together.  When  a  glass  rod  is  rubbed  with  silk  and  each  is  then  tested 
by  bringing  it  near  a  charged  electroscope,  it  is  found  that  the  glass 
has  a  positive  charge  and  the  silk  a  negative  one.  The  silk  being  a 
non-conductor,  retains  much  if  not  all  of  its  charge;  but  it  is  necessary 
to  proceed  somewhat  differently  if  we  wish  to  prove  that  the  charges 
are  equal.  It  is  also  better  to  use  vulcanite  and  flannel,  since  these 
materials  are  very  easily  electrified.  A  small  cap  of  flannel  is  made 
to  fit  over  the  end  of  the  rod, 
and  a  silk  thread  is  attached 
to  serve  as  an  insulating 
handle  (Fig.  370).  When  the 

end  of  the  rod  is  twisted  about  FIG.  370. 

in  the  cap,  and  is  then  brought 
near  a  charged  electroscope  with  the  cap  still  on  it,  it  produces  no 
effect;  but  when  the  cap  is  removed  by  means  of  the  thread,  a 
positive  charge  is  found  on  it  and  a  negative  charge  on  the  rod. 
Since  the  two  charges  exactly  neutralize  each  other  before  they  are 
separated,  they  must  be  equal  as  well  as  opposite. 


486    %  ELECTROSTATICS 

It  is  found  by  similar  tests  that  any  two  insulated  substances 
become  oppositely  electrified  when  rubbed  together,  and  that  the 
same  substance  receives  a  positive  charge  when  rubbed  with  cer- 
tain substances  and  a  negative  charge  when  rubbed  with  certain 
others;  e.g.  glass  is  positive  if  rubbed  with  silk,  but  negative  if  rubbed 
with  fur. 


393.  Electrical  machines  are  devices  for  producing  and 
collecting  electric  charges  more  conveniently  and  more  rap- 
idly than  is  possible  by  the  methods  already  described. 
They  are  of  two  types,  one  depending  upon  friction,  the 
other  upon  induction. 

One  form  of  friction  machine  is  shown  in  Fig.  371.    A 

positive  charge  is  de- 
veloped on  a  large 
revolving  glass  disk, 
A,  by  the  friction  of 
leather  pads,  B.  The 
charge  is  collected  on 
each  side  by  a  num- 
ber of  points  which 

FIG.  371.  -Friction  Machine.  from 


rod,  Fj  and  nearly  touch  the  disk.  The  rods  carry  the 
charge  to  an  insulated  brass  cylinder,  C,  from  which  it 
can  be  drawn  off  as  a  spark  discharge  by  bringing  the 
finger  or  any  other  conductor  near  it.  A  spark  a  centi- 
meter or  more  in  length  can  be  obtained  in  this  manner 
from  a  machine  in  good  condition.  Friction  machines  of 
various  forms  were  invented  during  the  eighteenth  century; 
but  they  are  greatly  inferior  to  the  more  modern  induc- 
tion machines,  and  are  no  longer  used  to  any  extent. 

The  simplest  and  earliest  form  of  induction  machine  is 
the  electrophorus  (Fig.  372).  It  consists  of  a  disk  of  vul- 
canite or  resinous  material,  and  a  metal  disk  or  cover"  of 


ELECTROSTATICS  487 

slightly  smaller  diameter,  provided  with  an  insulating 
handle.  The  vulcanite  is  negatively  electrified  by  strik- 
ing or  rubbing  it  with  cat's 
fur  or  flannel,  and  the  cover  is 
then  placed  upon  it.  Since 
the  vulcanite  is  a  non-con- 
ductor and  is  in  actual  con- 
tact with  the  metal  at  only 
a  few  points,  its  charge  does 
not  pass  to  the  cover.  But 
the  inductive  action  of  the 

charge  produces  an  opposite  or  positive  charge  on  the 
lower  side  of  the  cover,  and  a  negative  charge  on  the 
upper  side  (A,  Fig.  373).  The  negative  charge  is  repelled 
by  the  inducing  charge,  and  is  permitted  to  escape  by 
touching  the  cover  with  the  finger;  while  the  positive 


FIG.  373.  —  Action  of  the  Electrophorus. 

charge  is  retained  by  the  attraction  (5,  Fig.  373).  When 
the  cover  is  removed,  the  positive  charge  spreads  over  its 
entire  surface,  and  can  be  drawn  off  at  any  point  by  bring- 
ing the  finger  or  other  conductor  near  it.  Sparks  a  centi- 
meter long  can  be  obtained  in  this  manner.  The  cover  can 
be  repeatedly  charged  and  discharged  without  again  rub- 
bing the  vulcanite.  (Why?) 

Various  forms  of  induction  machines  have  been  invented, 
which  are  continuous  and  automatic  in  their  action  and  are 
much  more  powerful  than  the  electrophorus.  One  of  these, 


488 


ELECTROSTATICS 


called  the  Toepler-Holtz  machine,  is  shown  in  Fig.  374. 
It  has  a  revolving  glass  disk,  D,  and  a  stationary  one,  Df. 
To  both  are  attached  small  disks  and  strips  of  tin-foil,  F, 
which  serve  as  conductors.  While  the  machine  is  in  action, 
fixed  positive  and  negative  charges  accumulate  on  the 
metal-covered  parts  of  the  stationary  disk;  and  these  act 


FIG.  374.  —  Toepler-Holtz  Induction  Machine. 

inductively  on  the  metal  parts  of  the  revolving  disk, 
producing  a  positive  charge  on  one  side  of  the  axis  and  a 
negative  charge  on  the  other.  These  induced  charges  are 
collected  by  projecting  metallic  points,  as  in  the  friction 
machine,  and  accumulate  on  insulated  conductors  (rods, 
knobs,  and  Leyden  jars),  until  finally  a, spark  discharge 
occurs  across  the  air  space  between  them.  Sparks  from  5 
to  10  cm.  in  length  can  be  obtained  from  machines  of 
moderate  size.  Full  descriptions  of  these  machines  are 
to  be  found  in  larger  works  on  the  subject. 


ELECTROSTATICS  489 

394.  Distribution  of  a  Charge  on  a  Conductor.  —  Since  the  dif- 
ferent parts  of  an  electrical  charge  are  of  like  kind  and  repel  each  other, 
it  is  reasonable  to  suppose  that,  if  the  charged  body  is  a  conductor, 
this  mutual  repulsion  of  the  parts  will  drive  the  entire  charge  to  the 
outer  surface,  where  it  will  be  distributed  in  a  definite  manner  depend- 
ing on  the  shape  of  the  conductor.     Experiment  shows  that  this  is 
actually  the  case.     To  illustrate,  let  a  metal  vessel,  such  as  a  tin  can 
or  a  calorimeter,  be  placed  on  an  insulating  support  and  strongly 
charged  from  an  induction  machine.     When  a  proof  plane  (consisting 
of  a  small  metal  disk  with  an  insulating  handle)  is  touched  to  any  part 
of  the  outside  of  the  vessel  and  is  then  presented  to  an  electroscope, 
it  will  be  found  to  be  charged.     A  like  test  shows  that  the  proof  plane 
is  not  charged  by  contact  with  the  inside  of  the  vessel;  hence  the  entire 
charge  of  the  vessel  is  on  its  outer  surface. 

On  the  surface  of  a  spherical  conductor  a  charge  is  distributed  uni- 
formly; on  a  conductor  of  any  other  shape  the  distribution  varies 
with  the  curvature  of  the  different  parts  of  the  surface.     This  can  be 
shown    by    testing,    with    an    electroscope,    the 
strength  of  the  charge  received  by  a  proof  plane, 
when  touched  to  the  conductor  at  different  points. 
With  a  conductor  shaped  as  in  Fig.  375  it  will  be 
found  that   the  charge  is  greatest  at  A,  where 
the  curvature  is  greatest,  less  at  B,  and  least  at 
C,  where  the  curvature  is  least.      The   quantity 
of  electricity  per  unit  area  of  a  charged  body  is     FlG-  375 -—Insulated 

n    t     ,  -    ,        ,  T  Conductor. 

called  the  surface  density  of  the  charge.     In  gen- 
eral,  the  surface  density  on  a  charged  conductor  increases  as  we 
go   from  places   of  less  to   places   of  greater  curvature.     At  sharp 
projecting  points  the  electric  density  is  very  great. 

395.  Discharge  from  Points.  —  An  insulated  conductor  in  a  dry 
atmosphere  retains  a  charge  for  a  long  time,  provided  its  surface  is 
everywhere  smooth  and  gradually  curved;  but  at  any  sharp  point 
there  is  a  continuous  and  rapid  loss  of  the  charge  to  the  surrounding 
air.     The  nature  of  the  action  in  detail  is  a  theoretical  question 
which  need  not  be  considered  here.     The  net  result  is  that  the  charge 
passes  off  to  the  neighboring  air  particles,  which  are  then  repelled. 
These  charged  particles,  streaming  away  from  the  point,  form  a  cur- 
rent in  the  air,  known  as  an  "electrical  wind." 


490 


ELECTROSTATICS 


This  action  can  be  shown  by  attaching  a  pointed  wire  to  the  knob 
of  an  electrical  machine.     The  discharge  at  the  point  is  accompanied 

by  a  hissing  sound,  and,  in  a 
dark  room,  a  fine  jet  or  brush 
of  pale  blue  light  is  visible 
about  the  point.  The  wind 
can  be  felt  by  the  hand,  and 
the  flame  of  a  candle,  held 
near  the  point,  is  blown  aside 
(Fig.  376). 

A  conductor   may    be 
charged  as  well  as  discharged 


FIG.  376.  —  Electrical  Wind  Due  to 
Discharge  from  a  Point. 


by  the  action  of  points  on 
its  surface.  This  is  the  pur- 
pose of  the  rows  of  points 

which  extend  toward  the  revolving  disk  of  an  electrical  machine. 

The  charge  developed  on  the  disk  passes  across  through  the  air  to 

the  points. 

396.  Energy  of  a  Charge.  Electrical  Potential.  —  There 
is  a  definite  amount  of  energy  associated  with  every  elec- 
trical charge.  This  energy  is  manifested  in  various  ways. 
When  a  spark  discharge  occurs,  the  energy  of  the  charge 
is  converted  into  heat,  light,  and  sound.  The  heat  of 
even  a  short  spark  is  sufficient  to  light  a  gas  jet;  the  light 
and  sound  are  directly  evident  to  the  senses.  The  energy 
of  a  charge  is  further  shown  by  mechanical  effects.  For 
example,  a  piece  of  cardboard,  placed  between  the  knobs 
of  an  electrical  machine,  is  punctured  by  a  heavy  spark. 

The  energy  of  a  charge  is  a  form  of  potential  energy; 
and,  like  the  potential  energy  of  a  mass  raised  above  the 
earth,  its  value  is  determined  by  two  factors.  The  energy 
of  a  tank  of  water,  A  (Fig.  377),  standing  on  the  ground,  is 
measured  by  the  product  of  the  weight,  w,  of  the  water  and 
its  average  height.  If  the  height  of  the  surface  is  denoted 
by  h,  the  average  height  of  the  whole  body  of  water  is  ^  hy 


ELECTROSTATICS 


491 


and  its  potential  energy,  with  respect  to  the  level  of  the 
ground,  is  \  wh  foot-pounds.  If  the  water  is  drawn  off 
through  a  pipe  at  the  bottom  of  the  tank,  its  pressure  will 
steadily  decrease  from  a  maximum  at  the  start  to  zero,  as 
the  last  water  flows  out.  The  average  pressure  of  the  water 
is  half  the  pressure  at  the  start,  or  the  gravity  pressure  at 
half  the  original  depth.  We  may  just  as  well  take  this 
average  pressure  as  one  of  the  energy  factors,  instead  of 
the  average  height.  It  is  instructive  to  consider  the 

reverse  process.  When 
water  is  pumped  into  the 
tank  through  a  pipe  at  the 
bottom,  it  must  be  forced 


FIG.  377.  —  Electrical  Poten- 
tial Corresponds  to  Water 
Level. 

in  against  an  opposing 
pressure  which  steadily 
increases  as  the  level  of 

the  water  rises;  and  the  amount  of  work  required  to 
pump  in  each  succeeding  pound  of  water  increases  in  the 
same  ratio.  The  work  done  in  filling  the  tank  is  pro- 
portional to  the  average  value  of  this  pressure. 

These  energy  relations  are  exactly  paralleled  by  those 
involved  in  charging  and  discharging  a  conductor.  Sup- 
pose, for  example,  that  an  insulated  tin  can,  or  other  sim- 
ilar conductor,  is  charged  by  carrying  unit  quantities  of 
electricity  to  it  on  charged  pith  balls,  which  are  lowered 
within  and  touched  to  its  inner  surface.  As  each  new  por- 
tion of  the  charge  is  brought  up  from  a  distance,  it  is  re- 
pelled by  the  charge  already  on  the  conductor  (until  it  is 


4Q2  ELECTROSTATICS 

lowered  within),  and  work  must  be  done  in  overcoming 
this  repulsion.  The  repulsion  increases  in  proportion  to 
the  charge  on  the  conductor;  and  consequently  the  amount 
of  work  that  must  be  done  against  the  repulsion  in  bringing 
up  a  unit  charge  increases  in  the  same  ratio. 

The  work  done  against  the  repulsion  of  the  charge  in 
bringing  up  a  unit  quantity  of  electricity  to  it  is  called 
the  potential  of  the  charge,  or  the  electrical  potential  of  the 
conductor.  The  potential  increases  in  proportion  to  the 
charge,  as  stated  in  other  terms  just  above.  Since  it  is 
zero  when  the  charging  begins,  its  average  value  during 
the  process  is  half  its  final  or  maximum  value.  The 
product  of  this  average  potential  and  the  quantity  of  the 
charge  measures  the  total  work  done  against  the  repul- 
sion in  producing  the  charge;  and  this  is  also  the  energy 
of  the  charge.  Thus  the  two  factors  of  electrostatic 
energy  are  quantity  of  the  charge  and  potential  of  the  charge; 
and  these  correspond  respectively  to  weight  and  height  in 
the  case  of  the  tank  of  water.  Electrical  potential  may  also 
be  regarded  as  corresponding  to  fluid  pressure,  as  shown 
below;  indeed,  it  is  often  called  electrical  pressure. 

If  the  two  tanks  A  and  B  (Fig.  377)  are  connected  by  a 
pipe  at  the  bottom,  water  will  flow  from  A  to  B  until  the 
surface  stands  at  the  same  level  in  both.  It  is  difference 
of  level  that  determines  the  flow,  and  not  the  relative  quan- 
tities of  water  in  the  two  tanks.  Similarly,  when  two  posi- 
tively charged  conductors  are  connected  by  a  wire,  the  one 
at  the  higher  potential  loses  a  part  of  its  charge  to  the  other 
by  conduction  through  the  wire.  This  lowers  the  poten- 
tial of  the  first  conductor  and  raises  that  of  the  second  until 
they  become  equal.  Just  as  water  tends  to  flow  from  higher  to 
lower  levels,  so  positive  electricity  tends  to  flow  from  places 
at  higher  to  places  at  lower  electrical  potential. 


ELECTROSTATICS  493 

Elevation  is  measured  from  the  level  of  the  ground,  or, 
on  a  large  scale,  from  sea-level.  Taking  the  level  of  the 
ground  as  the  zero  of  elevation,  the  level  of  the  water  is 
positive  in  the  tanks  A  and  B  (Fig.  377)  and  negative  in 
the  wells  C  and  D.  If  the  wells  were  connected,  the  water 
would  flow  from  C  to  D,  i.e.  from  the  one  in  which  the 
negative  elevation  is  less  to  the  one  in  which  it  is  greater. 
Similar  relations  hold  between  positive  and  negative  po- 
tentials. The  electrical  potential  of  the  earth  is  taken  as 
zero.  The  earth  is  so  large  that  its  potential  is  not  mate- 
rially changed  by  any  positive  or  negative  charge  that  may 
be  imparted  to  it;  just  as  the  level  of  the  sea  is  not  percep- 
tibly changed  by  the  flow  of  rivers  into  it. 

Any  conductor  is  at  zero  potential  if,  when  electrically 
connected  with  the  earth,  there  is  no  flow  of  electricity 
from  either  to  the  other.  Its  potential  is  positive  if,  when 
thus  connected,  positive  electricity  flows  from  it  to  the 
earth,  and  negative  if  positive  electricity  flows  from  the 
earth  to  it.  A  positively  charged  body  has  a  positive 
potential,  a  negatively  charged  body  a  negative  potential, 
and  an  uncharged  body  a  zero  potential,  provided  there 
is  >  no  other  charged  body  in  its  vicinity.  The  greater 
the  positive  charge  on  a  body  the  higher  is  its  potential; 
the  greater  the  negative  charge  on  a  body  the  lower  is 
its  potential. 

As  long  as  the  level  of  the  water  in  the  tank  A  is  higher 
than  the  level  in  B,  the  pressures  at  the  two  ends  of  a  con- 
necting pipe  will  be  unequal;  and  the  difference  between 
these  pressures  is  the  immediate  cause  of  the  flow.  (When 
we  say  that  the  difference  of  level  is  the  cause  of  the  flow, 
we  go  one  step  farther  back  in  the  sequence  of  cause  and 
effect;  for  the  difference  of  pressure  is  due  to  the  difference 
of  level.)  Similarly  a  flow  of  positive  electricity  from 


494  ELECTROSTATICS 

higher  to  lower  potential  may  be  attributed  to  a  difference 
of  electrical  pressure,  acting  in  the  direction  of  the  flow. 

The  meaning  of  electrical  potential  is  presented  only  in  part  in 
the  above  discussion,  but  enough  has  been  said  to  serve  our  purpose. 
We  do  not  need  to  concern  ourselves  with  definitions  of  unit  charge, 
unit  potential,  and  the  various  other  electrostatic  units,  or  to  discuss 
the  methods  by  which  electrostatic  quantities  are  measured.  The 
study  of  electrostatics  is  principally  valuable  as  an  introduction  to 
the  study  of  electric  currents;  and  the  units  employed  in  the  latter 
branch  of  the  subject  are  different  from  the  electrostatic  units. 
We  may  anticipate  matters  by  saying  that  the  practical  unit  of 
potential  is  called  the  volt  and  when  potential  is  measured  in 
terms  of  this  unit,  it  is  called  voltage.  As  the  volt  is  more  or 
less  familiar  from  the  industrial  uses  of  the  electric  current,  we 
shall  adopt  it  in  advance. 

397.  Potential  and  Insulation.  Sparks.  —  As  a  gas  is 
pumped  into  a  closed  vessel  the  pressure  steadily  rises  until 
the  vessel  bursts  or  begins  to  leak.  This  maximum  pres- 
sure depends  on  the  strength  of  the  vessel,  and  not  on  its 
size.  Similarly,  as  the  charging  of  a  conductor  progresses, 
the  potential  rises,  until  finally  the  resisting  power  of  the 
air  or  other  insulating  medium  is  overtaxed,  and  the  charge 
either  leaks  off  gradually  or  escapes  suddenly  in  a  spark 
discharge.  The  potential  at  which  either  form  of  dis- 
charge occurs  depends  upon  the  strength  of  the  insula- 
tion, and  not  upon  the  material  of  the  conductor  or  to  any 
very  great  extent  upon  its  size.  (Shape  is  more  or  less 
important.  Recall  the  effect  of  sharp  points.) 

It  has  already  been  mentioned  that  the  value  of  differ- 
ent materials  as  insulators  depends  upon  the  particular 
use  to  which  they  are  put  (Art.  386).  It  will  now  be  under- 
stood that  the  one  determining  factor  is  the  potential  of 
the  electricity.  A  cotton  covering  is  quite  sufficient  to 
prevent  the  loss  of  electricity  from  wires  carrying  low- 


ELECTROSTATICS  495 

voltage  currents,  such  as  are  used  in  ringing  electric  bells; 
but  when  a  piece  of  this  wire  is  connected  with  an  induc- 
tion machine,  sparks  can  be  drawn  off  through  the  insula- 
tion as  readily  as  from  the  knob  of  the  machine.  Even  the 
heavy  cloth  and  rubber  covering  of  an  electric  light  wire 
makes  a  very  poor  showing  when  put  to  the  same  test; 
for  sparks  of  considerable  intensity  are  obtained  through  it. 
Such  tests  as  these  call  our  attention  again  to  the  fact 
that  only  the  strongest  insulation  is  effective  in  preventing 
the  escape  of  electric  charges.  (Note  that  the  charged 
parts  of  an  electrical  machine  are  all  insulated  by  several 
centimeters  of  vulcanite.)  Evidently  the  ordinary  poten- 
tials of  electric  charges  are  enormously  high  compared 
with  the  ordinary  potentials  of  electric  currents. 

The  potential  of  a  charged  conductor  can  be  roughly  estimated 
at  25,000  volts  for  each  centimeter  of  length  of  the  spark  that  passes 
between  it  and  a  second  conductor  at  zero  potential.  If  the  second 
conductor  is  also  charged,  the  length  of  spark  is  determined  by  the 
difference  of  potential  between  the  two.  A  potential  difference  of 
100,000  to  200,000  volts  between  the  knobs  of  an  induction  ma- 
chine is  not  uncommon.  In  fact,  the  potential  to  which  it  is  possible 
to  charge  the  machine  or  any  other  body  depends  only  upon  the 
strength  of  the  insulation  and  the  dryness  of  the  atmosphere.  Beyond 
this  limit  the  charge  escapes  as  rapidly  as  it  is  developed  or  imparted. 
Charges  at  even  the  highest  potentials  mentioned  are  not  danger- 
ous, unless  the  quantity  of  the  charge  is  much  larger  than  is  gener- 
ally the  case. 

398.  The  Leyden   Jar  and  Other  Condensers.  —  The 

Ley  den  jar  (Fig.  378)  is  a  device  for  accumulating  and  stor- 
ing a  large  charge.  Its  name  is  derived  from  the  city  of 
Leyden,  in  the  Netherlands,  where  the  principle  of  its 
action  first  became  known  in  1745.  It  consists  of  a  glass 
jar,  coated  inside  and  out  for  about  two  thirds  its  height 
with  tin-foil.  A  brass  rod,  terminating  in  a  knob  at  the 


496  ELECTROSTATICS 

top,  extends  through  the  cover,  and  is  connected  with  the 
inner  'coat  of  the  jar  by  means  of  a  chain,  attached  to  its 
lower  end.  To  charge  the  jar  it  is  held  in  the  hand,  and 
the  knob  is  brought  near  one  terminal  of  an  electrical 
machine;  or  it  may  be  placed  on  the  table,  and  the  knob 

connected  with  the  machine 
by  means  of  a  conductor.  In 
either  case  the  other  terminal 
of  the  machine  should  be  con- 
nected to  earth  by  running  a 
chain  or  wire  from  it  to  the 
table.  To  discharge  the  jar, 
FIG.  378.  —  Leyden  jar  and  one  end  of  a  short  conductor 

Discharger.  jg    touched    to    jts    Quter     CQat 

and  the  other  end  brought  near  the  knob,  as  shown  in 
the  figure.  Before  the  gap  is  closed  a  spark  passes,  dis- 
charging the  jar.  The  conductor  is  provided  with  an  in- 
sulating handle  to  protect  the  operator  from  the  danger 
of  a  shock. 

While  the  inner  coat  of  a  Leyden  jar  is  receiving  a  charge, 
the  outer  coat  is  also  receiving  one,  of  opposite  sign,  al- 
though its  potential  remains  at  zero.  The  latter  is  an 
induced  charge,  attracted  by  the  charge  on  the  inner  coat, 
and  is  received  from  the  earth,  by  conduction  through 
the  table  or  the  body  of  the  person  holding  the  jar.  This 
induced  charge  reacts  inductively  on  the  inner  one,  and  by 
its  attraction  enables  the  inner  coat  to  receive  a  much 
greater  charge  from  the  machine  than  would  otherwise  be 
possible.  To  prove  this  we  have  only  to  charge  and  dis- 
charge the  jar  while  it  is  standing  on  a  large  sheet  of  vul- 
canite or  glass.  With  the  outer  coat  thus  insulated,  it 
can  not  become  charged,  and  the  jar  can  be  made  to  yield 
only  a  short,  weak  spark. 


ELECTROSTATICS  497 

The  extent  to  which  the  electrical  capacity  of  a  Leyden  jar  is 
increased  by  the  mutual  induction  of  the  opposite  charges  on  its  inner 
and  outer  coats  is  strikingly  shown  by  the  action  of  the  jars  of  an  in- 
duction machine  (Fig.  374).  When  the  machine  is  operated,  the 
charges  accumulate  principally  in  the  jars,  the  positive  charge  in  one, 
the  negative  in  the  other.  The  outer  coats  of  the  jars  become  oppo- 
sitely charged  by  induction,  each  receiving  its  charge  from  the  other 
through  metal  conducting  rods,  by  which  they  may  be  connected 
at  the  will  of  the  operator.  Under  these  conditions  the  machine 
gives  a  thick,  brilliant  spark,  at  intervals  of  several  seconds;  but  when 
the  outer  coats  of  the  jars  are  disconnected,  the  sparks  are  thin  and 


FIG.  379.  —  Photograph  of  a  Lightning  Flash. 

r 

faint,  and  occur  much  more  frequently.  The  quantity  of  electricity 
that  is  discharged  with  each  spark  is  very  much  less  in  the  latter  case 
owing  to  the  reduced  capacity  of  the  jars.  The  loss  of  capacity  is 
due  to  the  fact  that  the  outer  coats  of  the  jars  do  not  become  charged, 
the  wooden  base  of  the  machine  being  practically  an  insulator  for 
such  rapid  action. 

399.  Atmospheric  Electricity.  —  The  sparks  obtained 
from  electrical  machines  and  Leyden  jars  suggested  to  a 
number  of  the  early  experimenters  in  electricity  that 
lightning  was  an  electrical  discharge  between  one  cloud 
and  another  or  between  a  cloud  and  the  earth.  Benjamin 


498  ELECTROSTATICS 

Franklin  put  this  theory  to  an  experimental  test  in  1752. 
"He  sent  up  a  kite  during  the  passing  of  a  storm,  and 
found  the  wetted  string  to  conduct  electricity  to  the  earth, 
and  to  yield  an  abundance  of  sparks.  These  he  drew  from 
a  key  tied  to  the  string,  a  silk  ribbon  being  interposed 
between  his  hand  and  the  key  for  safety.  Leyden  jars 
could  be  charged,  and  all  other  electrical  effects  produced, 
by  the  sparks  furnished  from  the  clouds.  The  proof  of 
the  identity  was  complete." 

Thunder  corresponds  to  the  snapping  sound  produced 
by  an  electric  spark.  The  sudden  heating  of  the  air  along 
the  path  of  a  lightning  flash  causes  it  to  expand  with  explo- 
sive violence,  producing  sound  waves  of  great  intensity. 
If  the  flash  is  short  and  straight,  the  sound  is  a  short  clap; 
if  it  is  long  and  zigzag,  the  sound  produced  by  its  differ- 
ent parts  have  unequal  distances  to  travel  to  the  observer 
and  are  heard  in  quick  succession  as  a  continuous  rattle. 
The  rolling  sound  of  distant  thunder  is  due  to  various 
reflections  of  the  sound  from  clouds,  from  the  ground,  and 
often  from  neighboring  hills. 

Experiments  have  shown  that  the  atmosphere  is  gen- 
erally electrified  even  in  fair  weather.  In  fair  weather  the 
electrification  is  almost  always  positive;  in  stormy  weather 
it  is  sometimes  positive  and  sometimes  negative.  The 
potential  increases  with  the  altitude;  but  differs  widely  in 
different  localities  and  with  different  states  of  the  weather. 
The  rise  of  potential  has  been  found  as  great  as  600  volts 
per  meter  of  elevation  above  the  ground.  The  potential 
of  thunder-clouds,  as  estimated  from  the  length  of  light- 
ning flashes,  runs  into  the  hundreds  of  millions  of  volts. 
The  aurora  or  northern  light  is  due  to  electric  discharges 
in  the  upper  air.  (Art.  500.) 

Various  theories  have  been  advanced  to  account  for  the 


ELECTROSTATICS  499 

electrification  of  the  atmosphere;  but  very  little  is  defi- 
nitely known  about  it.  Evaporation  is  very  probably  one 
of  the  principal  causes.  « 

400.  Lightning  Conductors.  —  The  use  of  lightning  conductors 
to  protect  buildings  was  first  suggested  by  Benjamin  Franklin.     The 
usual  device  consists  of  one  or  more  iron  rods,  extending  some  dis- 
tance above  the  highest  points  of  the  building  and  connected  by  means 
of  large  iron  or  copper  conductors  with  damp  earth,  or,  better,  with 
water.    A  conductor  ending  in  dry  earth  is  worse  than  useless,  it 
is  dangerous;  for  dry  earth  is  not  a  sufficiently  good  conductor.     Each 
rod  is  terminated  by  a  gilded  copper  point. 

The  action  of  a  lightning  conductor  depends  largely  upon  induc- 
tion. A  charged  cloud  induces  an  opposite  charge  at  the  surface  of 
the  ground  under  it  and  on  houses,  trees,  and  other  objects  within 
this  area.  The  inductive  action  is  strongest  upon  the  highest  objects, 
and  causes  lightning  rods  to  become  highly  electrified.  Under  these 
conditions  a  rapid  and  continuous  brush  discharge  takes  place  from 
the  sharply  pointed  tips  of  the  rods.  This  quiet  discharge  of  oppo- 
site electrification  toward  the  cloud  is  often  sufficient  to  prevent  light- 
ning; but,  if  a  stroke  does  occur,  it  is  received  by  the  rod  and  the 
building  is  not  damaged. 

401.  The  Electric  Field.  —  The  attraction  or  repulsion  between 
two  electric  charges  varies  in  amount  with  the  intervening  medium. 
For  example,  it  is  one  sixth  as  great  through  glass  as  it  is  through  an 
equal  thickness  of  air.     But  electric  forces  act  as  readily  through  a 
vacuum  as  they  do  through  air;  from  which  it  appears  that  the  ether 
is  the  one  essential  medium,  as  it  is  in  the  case  of  magnetic  forces. 
Magnetic  action  and  electrostatic  action  are,  however,  fundamentally 
different  in  their  nature;  for  an  electric  charge  neither  attracts  nor 
repels  a  magnetic  pole. 

The  space  within  which  an  electric  charge  can  be  detected  is  called 
an  electric  field ;  and  a  line  of  electric  force  is  a  line  in  the  field  along 
which  an  electric  charge  would  move,  under  the  attraction  or  repul- 
sion of  the  field. 

The  energy  of  a  charge  is  stored,  not  on  or  in  the  conductor, 
but  in  the  electric  field,  and  is  due  to  a  state  of  strain  in  the  ether 
and  other  insulating  media  which  occupy  the  field.  This  is  proved 
by  the  occurrence  of  electric  sparks,  which  are  due  to  the  breaking 


500  ELECTROSTATICS 

down  of  the  material  structure  of  the  medium.  "If  a  spark  passes 
through  a  sheet  of  paper  or  a  pane  of  glass,  a  hole  is  made  in  it;  if  the 
spark  is  in  air,  the  molecules  of  its  gases  are  broken  into  parts.  This 
proves  that  the  medium  must  have  been  greatly  strained  just  before 
the  sparks  passed;  and,  if  it  was  strained,  it  must  have  possessed 
potential  energy." 

Although  the  ether  transmits  electric  force,  it  does  not  transmit 
electricity.  A  perfect  vacuum  is  a  perfect  insulator,  and  a  spark  in 
it  is  impossible. 


CHAPTER  XIII 
ELECTRODYNAMICS 

I.    INTRODUCTION 

402.  Effects  of  the  Electric  Current.  —  An  electric  cur- 
rent can  not  be  seen,  but  its  presence  is  known  by  the  effects 
which  it  produces.  Some  of  the  effects  of  high-potential 
currents  are  already  familiar,  such  as  the  electric  spark 
and  the  shock  experienced  when  the  current  passes  through 
any  part  of  the  body.  Other  effects  are  produced  by  cur- 
rents of  both  high  and  low  potential.  These  are  classed 
as  magnetic  effects,  heating  effects,  and  chemical  effects. 

The  heating  effect  is  well  known  through  its  application 
in  electric  lighting.  It  is  an  obvious  fact  that  the  light  of 
an  arc  or  an  incandescent  lamp  comes  from  a  white-hot 
body.  Small  incandescent  lamps  are  made  which  are  bril- 
liantly lighted  by  the  current  from  a  battery  of  three  or 
four  dry  cells. 

Magnetic  effects  are  also  familiar,  but  their  nature  is 
less  evident.  One  may  ring  door  bells  and  use  telephones, 
and  yet  remain  in  ignorance  of  the  fact  that  these  useful 
appliances  owe  their  existence  to  the  magnetic  field  which 
surrounds  an  electric  current.  The  existence  of  such  a 
field  is  shown  by  the  deflection  of  a  magnetic  needle,  when 
near  a  wire  in  which  a  current  is  flowing.  If  the  wire  is 
extended  parallel  to  the  needle,  at  a  distance  of  several 
centimeters  above  it,  and  is  then  brought  down  close  to  it 
(Fig.  380),  the  needle  will  be  deflected  through  a  greater  or 
less  angle,  depending  upon  the  strength  of  the  current. 

501 


502 


ELECTRODYNAMICS 


We  have  here  a  most  important  difference  between  elec- 
tricity at  rest  and  electricity  in  motion;  when  in  motion 
it  acts  upon  magnets,  when  at  rest  it  does  not. 

The  chemical  effects 
of  electric  currents  are 
less  familiar,  but  of 
great  and  growing  im- 
portance in  the  chemical 
industries.  One  of  the 
simplest  examples  of 

FIG.  380.  —  Magnetic  Action  of  a  Current.  .          .      . 

electrochemical  action  is 

the  separation  of  water  into  its  constituent  gases,  hydrogen 
and  oxygen,  when  a  current  is  passed  through  water  con- 
taining a  little  sulphuric  acid. 

In  the  study  of  electric  currents  we  shall  become  ac- 
quainted with  these  effects  in  detail,  and  with  many  of 
their  more  important  applications. 

403.  Sources  of  Electric  Currents.  —  Electric  currents 
for  all  practical  purposes  are  generated  either  by  electric 
batteries  or  by  dynamos.  Batteries  are  used  where  the  cur- 
rent required  is  comparatively  small,  as  in  ringing  bells, 
operating  telegraph  instruments,  etc.  Large  ," storage" 
batteries  generate  electricity  in  sufficient  quantity  to  run 
electric  automobiles  and  electric  launches;  but  they  must 
be  " charged"  at  frequent  intervals  by  currents  from  dyna- 
mos. Dynamos  are  thus  the  only  primary  or  original 
source  of  electrical  energy  on  a  large  scale. 

It  might  be  supposed  that  an  induction  machine  would  be 
capable  of  supplying  a  considerable  current,  comparable, 
at  least,  with  the  current  from  ordinary  batteries;  but  a 
simple  test  proves  that  this  is  not  the  case.  A  small  elec- 
tric bell  can  be  rung  by  means  of  a  single  electric  cell  of 


PRIMARY  CELLS  503 

almost  any  type;  but,  when  the  bell  is  connected  with  the 
knobs  of  an  induction  machine,  it  remains  silent,  however 
vigorously  the  machine  may  be  operated.  The  relative 
strength  of  the  currents  is  more  definitely  shown  by  their 
action  on  a  magnetic  needle,  when  flowing  through  a  wire 
directly  above  it  (Fig.  380).  The  current  from  a  cell 
deflects  the  needle  several  degrees  —  perhaps  20°  or  30°. 
If  we  connect  the  ends  of  a  wire  with  the  knobs  of  an  in- 
duction machine,  the  current  from  the  machine  flows 
through  it ;  but  the  needle  is  not  deflected  when  the  wire 
is  brought  near  it.  Since  the  magnetic  action  is  too  weak 
to  affect  the  needle,  the  current  must  be  very  small. 

II.  PRIMARY  CELLS 

404.  General  Facts  Concerning  Electric  Cells.  —  There 
are  many  forms  of  electric  cells,  but  they  are  all  alike  in 
certain  respects.  Every  cell  has  two  plates.  In  most 
cells  one  plate  is  of  zinc  and  the  other  of  copper  or 'carbon. 
Every  cell  contains  a  liquid  in  which  the  plates  are  im- 
mersed, or  two  liquids,  with  one  of  the  plates  immersed  in 
each.  (In  the  so-called  dry  cell  the  liquid  is  held  by  a 
porous  solid.)  Different  liquids  are  used,  a  common  one 
being  dilute  sulphuric  acid.  To  obtain  a  current  from  a 
cell  a  wire  is  connected  with  its  plates.  For  convenience 
in  making  the  connection,  each  plate  has  a  binding  post 
at  the  top. 

A  cell  in  working  condition  supplies  a  current  continu- 
ously, as  long  as  its  plates  are  connected  by  a  conductor. 
Now  we  have  learned  in  the  study  of  electrostatics  that  a 
difference  of  potential  is  a  necessary  condition  for  the  flow 
of  electricity.  Hence  we  may  reasonably  infer  that  the 
plates  of  a  cell  are  at  different  potentials.  The  difference 
is  very  small,  however,  and  can  be  detected  only  with 


504 


ELECTRODYNAMICS 


a  very  sensitive  apparatus.     For  this   purpose  the  elec- 
troscope must  be  provided  with  two  metal  disks,  which 

are  covered  on  their 
contact  surfaces  with 
an  insulating  coat  of 
shellac,  and  act  as  con- 
densers (Fig.  381).  One 
disk  takes  the  place 
of  the  customary  knob 
of  the  electroscope,  and 


FIG.  381. —  Demonstrating  that  the  Plates  of 
a  Cell  are  Charged. 


the  other  is  provided 
with  an  insulating 
handle.  To  make  the  test  the  disks  are  placed  together, 
and  each  is  connected  by  a  wire  with  a  plate  of  the  cell,  as 
shown  in  the  figure.  The  wires  are  then  removed  and  the 
upper  disk  lifted  off.  The  charge  on  the  disk  of  the  electro- 
scope is  now  free,  and  is  shared  with  the  leaves,  causing  them 
to  diverge  slightly.  When  this  charge  is  tested  in  the  usual 
manner,  it  proves  to  be  negative  if  it  was  received  from  the 
zinc  plate,  and  positive  if  received  from  the  copper  or  the 
carbon  plate  of  the  cell.  If  stronger  charges  are  desired 
for  the  test,  they  may  be  obtained  from  a  battery  of  two 
or  more  cells  connected  in  series. 

When  the  plates  of  a  cell  are  joined  by  a  wire,  positive 
electricity  flows  through  the  wire  from  the  positively  charged 
copper  or  carbon  plate  to  the  negatively  charged  zinc, 
i.e.  from  positive  to  negative  potential.  This  is  called 
the  direction  of  the  current.  There  is  an  equal  flow  of 
negative  electricity  in  the  opposite  direction;  but  a  nega- 
tive current  is  equivalent  in  its  effects  to  an  equal  positive 
current,  and  it  is  the  universal  practice  to  regard  the  entire 
current  as  positive.  (The  newer  theory — Art.  512 — gives 
a  different  account  of  the  process.) 


PRIMARY   CELLS  505 

The  plates  of  a  cell  are  often  called  poles  or  electrodes 
(from  the  Greek  electro  +  hodos,  way,  i.e.  a  way  for  elec- 
tricity). The  copper  or  carbon  is  called  the  positive  pole 
or  electrode,  and  the  zinc  the  negative  pole  or  electrode. 

405.  Further  Comparison  of  Electric  Cells  and  the  In- 
duction Machine.  —  The  flow  of  water  through  a  pipe 
increases  with  an  increased  difference  between  the  pressures 
at  the  inlet  and  the  outlet;  so,  too,  the  flow  of  electricity 
through  a  given  conductor  increases  with  an  increased 
potential  difference  between  its  terminals.  This  being  the 
case,  it  might  be  supposed  that  an  induction  machine,  which 
can  be  charged  to  a  potential  difference  of  100,000  volts, 
would  send  a  much  larger  current  through  a  wire  than  an 
electric  cell,  the  plates  of  which  differ  in  potential  by 
one  or  two  volts  at  the  most.  But  we  have  seen  that 
the  current  from  the  cell  is  much  the  greater.  This 
apparent  contradiction  is  explained  when  we  take  account 
of  all  the  facts. 

When  the  knobs  of  an  electrical  machine  are  connected 
by  a  wire,  no  spark  can  be  drawn  from  the  machine,  not 
even  the  shortest,  and  no  shock  is  felt  when  the  knobs 
are  touched  with  the  fingers.  The  positive  and  negative 
electricities  flow  through  the  wire  and  neutralize  each  other 
as  fast  as  they  are  developed,  and  no  accumulation  of 
charges  is  possible.  It  is  like  pouring  water  into  a  sieve. 
However  industriously  this  may  be  done,  the  sieve  remains 
empty.  In  like  manner  the  knobs  and  jars  of  the  machine 
remain  practically  at  zero  potential.  We  see,  then,  that 
if  a  cell  can  maintain  a  potential  difference  of  one  volt, 
or  even  a  small  fraction  of  a  volt,  between  its  poles,  it  is 
superior  to  the  induction  machine  as  a  source  of  a  con- 
tinuous flow  of  current. 


506  ELECTRODYNAMICS 

When  an  induction  machine  is  operated  in  the  usual 
manner  and  a  spark  passes,  there  is  a  sudden  rush  of  cur- 
rent, which  lasts  perhaps  for  a  millionth  of  a  second.  This 
reduces  the  potential  difference  to  zero,  and  it  must  again 
become  very  great  before  another  spark  discharge  can  take 
place.  The  time  intervals  between  successive  sparks 
are  very  great  compared  with  the  actual  duration  of  the 
current;  hence  the  quantity  of  electricity  that  passes  per 
second  is  exceedingly  small.  A  cell,  on  the  contrary, 
maintains  a  steady  flow  of  current;  from  which. we  know 
that  the  plates  must  be  recharged,  by  some  action  within 
the  cell,  as  rapidly  as  they  are  discharged  through  the  wire. 

The  charged  plates  of  a  cell  possess  energy,  which  is 
constantly  expended  in  maintaining  the  current  and  con- 
stantly renewed  by  the  action  within  the  cell.  Evidently 
there  must  be  a  certain  store  of  energy  in  the  cell,  which 
is  available  for  this  work.  The  source  of  this  energy  and 
something  of  the  manner  in  which  it  is  liberated  may  be 
learned  from  a  study  of  the  earliest  and  simplest  form  of 
electric  cell,  invented  by  the  noted  Italian  physicist,  Ales- 
sandro  Volta,  in  1800.  It  is  named  after  him  the  voltaic 
cell,  and  consists  simply  of  a  zinc  and  a  copper  strip  in 
dilute  sulphuric  acid.  To  explain  the  action  of  the  cell,  we 
must  begin  with  the  chemical  behavior  of  these  materials. 

406.  Action  of  Dilute  Sulphuric  Acid  on  Zinc  and  Cop- 
per. —  When  a  strip  of  zinc  is  placed  in  dilute  sulphuric 
acid,  it  is  attacked  by  the  acid  and  gradually  eaten  away 
or  dissolved.  At  the  same  time  small  bubbles  form  in 
great  numbers  on  the  surface  of  the  zinc,  to  which  they 
adhere  until  detached  by  the  buoyancy  of  the  liquid. 
The  escape  of  the  bubbles  at  the  surface  gives  the  liquid 
the  appearance  of  boiling. 


PRIMARY  CELLS  507 

These  visible  effects  are  due  to  chemical  action.  The 
sulphuric  acid  molecule  consists  of  two  atoms  of  hydro- 
gen, one  of  sulphur,  and  four  of  oxygen,  and  is  represented 
by  the  formula  H2S04.  In  the  chemical  action  the  two 
hydrogen  atoms  of  the  acid  molecule  are  replaced  by  one 
atom  of  zinc  (Zn),  forming  a  molecule  of  zinc  sulphate 
(ZnSO4);  and  the  two  hydrogen  atoms  unite  to  form  a 
hydrogen  molecule.  The  hydrogen  molecules  gather  in 
the  form  of  bubbles  and  escape.  The  zinc  sulphate  re- 
mains in  solution  in  the  liquid.  If  the  liquid  is  evaporated, 
the  sulphate  remains  in  the  form  of  a  white  solid. 

As  the  action  of  the  acid  on  the  zinc  continues,  the  tem- 
perature of  the  liquid  rises.  Evidently  heat  is  generated 
in  the  process.  The  acid  and  the  zinc  possess  a  certain 
amount  of  chemical  energy,  which  is  converted  into  heat 
when  the  two  substances  unite.  This  transformation  of 
energy  is  similar  to  that  which  takes  place  when  fuel  is 
burned.  The  zinc  may  be  compared  to  coal  and  the  acid 
to  the  oxygen  of  the  air,  with  which  the  coal  unites  in 
burning. 

When  a  strip  of  copper  is  placed  in  the  acid,  no  bubbles 
are  formed  and  the  copper  does  not  waste  away  however 
long  it  may  remain  in  the  liquid.  There  is  no  appreciable 
chemical  action. 

407.  Electrochemical  Action  in  the  Voltaic  Cell.  —  When 
a  zinc  and  a  copper  strip  are  in  the  same  vessel  of  dilute 
sulphuric  acid,  but  are  not  in  contact,  neither  strip  is  af- 
fected by  the  presence  of  the  other,  and  hydrogen  bubbles 
appear  only  on  the  zinc.  When  the  strips  are  connected 
by  a  wire,  hydrogen  bubbles  form  on  both,  and  a  mag- 
netic needle  indicates  the  presence  of  an  electric  current 
in  the  wire  (Fig.  380). 


508  ELECTRODYNAMICS 

While  the  current  is  flowing  the  acid  appears  to  attack 
both  strips,  since  bubbles  form  on  both;  but  the  appear- 
ance deceives.  The  copper  does  not  waste  away  however 
long  it  may  be  used.  The  hydrogen  liberated  at  the  copper 
represents  useful  consumption  of  zinc  and  acid.  By  this 
action  the  strips  are  charged  and  the  current  is  maintained 
in  the  wire.  The  hydrogen  liberated  at  the  zinc  repre- 
sents useless  consumption  of  zinc  and  acid,  by  which  the 
chemical  energy  of  the  materials  is  immediately  converted 
into  heat  in  the  liquid.  This  useless  action  takes  place 
whether  the  cell  is  generating  a  current  or  not.  The  waste- 
ful action  may  be  greatly  reduced  by  treating  the  zinc 
with  mercury  (Art.  410). 

The  electrical  nature  of  the  action  in  the  cell  remains  to  be  con- 
sidered. This  is  explained  by  the  theory  of  electrolytic  dissociation. 
According  to  this  theory  many  substances,  which  are  known  in  chem- 
istry as  acids,  bases,  and  salts,  become  more  or  less  dissociated  when 
they  are  dissolved  in  water.  A  dissociated  molecule  is  one  that  is 
broken  up  into  two  or  more  electrified  parts,  called  ions.  An  ion  is 
an  atom  or  a  group  of  atoms  having  a  positive  or  a  negative  charge. 
Since  the  solution  as  a  whole  is  not  charged,  the  sum  of  all  the  charges 
on  the  positive  ions  must  be  equal  to  the  sum  of  all  the  charges  on 
the  negative  ions.  The  sulphuric-acid  molecule  forms  two  positive 
hydrogen  ions  (H+,H+)  and  one  negative  ion  (SO")-  The 
latter  is  called  a  sulphion.  Its  negative  charge  is  equal  to  the  sum  of 
the  positive  charges  on  the  two  hydrogen  ions. 

When  a  strip  of  zinc  is  placed  in  the  acid,  it  immediately  begins  to 
dissolve  by  giving  off  positively  charged  zinc  ions  (Zn++)  to  the 
liquid.  The  charge  on  the  zinc  ion  is  twice  as  great  as  that  on  the 
hydrogen  ion.  The  loss  of  these  positive  charges  leaves  the  zinc 
negatively  charged;  while  the  liquid  immediately  surrounding  the 
zinc  plate  is  positively  charged  by  the  presence  of  the  zinc  ions. 
These  charges  quickly  increase  to  a  definite  limit;  for  a  zinc  ion,  on 
the  point  of  leaving  the  plate  with  its  positive  charge,  is  retarded 
by  the  attraction  of  the  negative  charge  on  the  plate  and  also  by 
the  repulsion  of  the  positive  charge  of  the  liquid.  These  retarding 


PRIMARY   CELLS 


509 


forces  soon  become  great  enough  to  prevent  the  further  escape  of 
the  zinc  ions  into  the  liquid,  unless  the  charges  are  carried  off  in 
some  way  as  they  accumulate. 

A  copper  strip,  placed  in  the  acid  and  connected  by  a  wire  with 
the  zinc,  completes  an  electric  circuit  through  which  the  discharge 
can  take  place.  The  positive  zinc  and  hydrogen  ions  are  repelled 
from  the  space  about  the  zinc  plate,  where  the  potential  of  the  liquid 
is  the  highest,  and  drift  toward  the  copper  plate.  On  arriving  at 
the  copper  plate,  the  hydrogen  ions  give  up  their  positive  charges  to 
it,  and  unite  in  pairs,  forming  uncharged  hydrogen  molecules.  These 
accumulate  in  the  form  of  bubbles  and  escape.  The  positive  charge 
on  the  copper  is  conducted  through  the  wire  to  the  zinc  plate.  This 
raises  the  potential  of  the  zinc  and  enables  it  to  give  off  more  ions. 
The  current  is  thus  maintained  as  long  as  the  plates  are  connected, 
or  until  the  zinc  is  entirely  dissolved  or  the  supply  of  hydrogen  ions 
is  exhausted. 


408.   Mechanical  Illustration  of  the  Action  of  a  Cell.  - 

We  have  already  made  use  of  mechanical  analogies  in  dis- 
cussing electrical  phenomena,  and  fur- 
ther helpful  ideas  may  be  gained  in 
the  same  way.  The  action  of  a  cell  in 
producing  and  maintaining  a  difference 
of  potential  may  be  compared  to  the 
action  of  a  pump  in  producing  and  main- 
taining a  difference  of  water-level.  With 
the  device  shown  in  Fig.  382,  water  can 
be  forced  through  the  lower  pipe  from 
L  .to  R,  by  means  of  the  rotary  pump, 
which  is  driven  by  the  weight  W.  The 
back  pressure  of  the  water  in  R  increases 
with  the  increasing  difference  of  the 
water-levels  in  R  and  L  until,  finally,  it 
becomes  great  enough  to  stop  the  pump.  If  now  the  stop- 
cock in  the  upper  pipe  is  opened,  water  will  flow  through 


FIG.  382.  — Water 
Analogy  of  an 
Electric  Cell. 


510  ELECTRODYNAMICS 

it  from  R  to  L,  the  level  in  R  will  fall  somewhat,  thus 
decreasing  the  back  pressure  against  the  pump,  and  the 
pump  will  start  again. 

Under  these  conditions  there  will  be  a  continuous  circu- 
lation of  the  water,  the  pump  supplying  the  energy  expended 
in  maintaining  the  flow. 

In  this  illustration  the  tank  R  represents  the  copper 
plate,  L  the  zinc  plate,  and  the  action  of  the  pump  the 
chemical  action  in  the  cell.  The  upper  pipe,  with  the  stop- 
cock open,  represents  the  connecting  wire  between  the 
plates.  When  the  stop-cock  is  closed,  the  return  flow  is 
cut  off;  and,  as  soon  as  the  difference  of  level  in  R  and  L 
reaches  the  possible  maximum,  the  pump  stops.  Similarly, 
when  the  plates  of  a  cell  are  disconnected,  their  charges 
accumulate  until  the  potential  difference  reaches  its  maxi- 
mum value  for  that  particular  type  of  cell.  All  action 
within  the  cell  then  ceases,  unless  there  is  wasteful  consump- 
tion of  the  zinc. 

The  maximum  potential  difference  which  a  cell  is  able 
to  produce,  i.e.  the  difference  between  the  potentials  of 
its  plates  when  they  are  disconnected,  is  called  the  electro- 
motive force  of  the  cell.  Potential  difference  is  usually 
denoted  by  P.D.  and  electromotive  force  by  E.M.F.  The 
E.M.F.  of  the  zinc-copper-sulphuric-acid  cell  is  approxi- 
mately one  volt. 

409.  The  Electric  Circuit.  —  When  the  plates  of  a  cell 
are  connected  by  a  wire,  the  positive  electricity  flows 
through  the  liquid  from  the  zinc  to  the  copper  plate,  thence 
through  the  wire  to  the  zinc  plate.  Its  path  is  a  complete 
circuit,  continuous  from  any  point  back  to  that  point 
again.  In  general,  an  electric  circuit  consists  of  a  series 
of  conductors,  forming  a  closed  loop.  A  circuit  is  said  to 


PRIMARY   CELLS  511 

be  closed  when  it  is  complete,  open  or  broken  when  there 
is  a  gap  at  any  point. 

Electricity  flows  with  a  velocity  comparable  with  that 
of  light,  and  the  current  is  established  in  all  parts  of  a 
circuit  practically  at  the  same  instant. 

When  we  say  that  a  cell  "  generates  an  electric  current," 
the  expression  is  to  be  understood  to  mean  that  the  action 
in  the  cell  establishes  and  maintains  a  flow  of  electricity. 
Before  the  circuit  is  closed  the  electricity  is  already  in 
existence,  in  the  form  of  charges  carried  by  the  ions  in  the 
liquid  and  charges  on  the  plates.  In  the  same  sense  we 
may  say  that  a  pump  generates  a  current  of  water.  Just 
as  the  same  body  of  water  may  be  caused  to  flow  endlessly 
round  a  circuit  (Fig.  382),  conveying  water-power  from  one 
point  to  another  without  loss  of  the  water  itself,  so  the  same 
electricity  may  be  caused  to  flow  endlessly  round  a  circuit, 
conveying  electrical  power,  and  this  power  can  be  used  in 
ringing  bells,  lighting  lamps,  running  motors,  etc.,  without 
loss  of  the  electricity  itself.  Some  of  the  water  may  be 
lost  through  leaky  pipes,  and  some  of  the  electricity  may 
be  lost  through  poor  insulation;  but  neither  of  these  losses 
is  a  necessary  or  useful  part  of  the  process. 

410.  Local  Action  on  the  Negative  Plate. — We  have  seen 
that  hydrogen  is  liberated  at  the  zinc  plate  of  a  voltaic 
cell  whether  the  circuit  is  open  or  closed,  and  have  learned 
that  this  represents  a  wasteful  consumption  of  the  zinc 
and  acid.  This  action  is  due  to  small  particles  of  iron, 
lead,  and  carbon,  which  are  present  as  impurities  in  com- 
mercial zinc.  Any  such  particle  on  the  surface  of  the  zinc 
and  in  contact  with  the  liquid  acts  as  a  positive  pole,  and 
forms  a  minute  voltaic  cell  with  the  adjacent  zinc  and 
liquid.  This  causes  a  local  or  parasitic  current  at  the 


5 1 2  ELECTRODYNAMICS 

spot,    which    adds    nothing    to    the    flow    through    the 
wire. 

With  chemically  pure  zinc  this  local  action,  as  it  is  called, 
does  not  occur.  The  zinc  is  consumed  only  when  the  cir- 
cuit is  closed,  and  the  hydrogen  is  set  free  only  at  the  posi- 
tive plate.  The  same  result  is  obtained,  though  somewhat 
imperfectly,  with  a  plate  of  commercial  zinc,  when  covered 
with  a  coating  of  mercury.  The  mercury  dissolves  a  por- 
tion of  the  zinc,  forming  a  pasty  amalgam,  which  covers 
the  surface  and  keeps  the  acid  from  contact  with  the  impu- 
rities. The  zinc  in  this  condition  is  said  to  be  amalgamated. 
Amalgamation  prevents  a  great  deal  of  waste  where  the 
liquid  of  the  cell  is  an  acid  solution,  as  in  the  bichromate 
cell  (Art.  413).  With  the  various  types  of  cells  in  general 
use,  the  materials  used  in  the  liquid  are  such  that  local 
action  is  avoided,  and  amalgamation  is  unnecessary. 

411.  Polarization  of  the  Positive  Plate.  —  The  simple 
voltaic  cell  has  another  defect,  due  to  the  accumulation  of 
hydrogen  on  the  positive  plate.  In  addition  to  the  bubbles, 
a  thin,  invisible  film  of  hydrogen  spreads  over  the  surface. 
The  result  is  a  very  considerable  weakening  of  the  current. 
This  may  be  shown  with  a  battery  of  one  or  more  cells, 
which,  when  first  connected  in  circuit,  is  just  sufficient  to 
operate  a  telegraph  sounder.  The  current  very  quickly 
becomes  too  weak  for  the  purpose.  The  power  of  the 
battery  can  be  restored  by  drying  the  copper  plates  in  a 
Bunsen  flame. 

A  deposit  of  hydrogen  on  the  positive  plate  of  a  cell 
decreases  the  current  for  two  reasons.  In  the  first  place, 
hydrogen  is  a  non-conductor,  and  cuts  off  the  current  from 
the  part  of  the  surface  that  it  covers.  The  cell  as  a  whole 
thus  becomes  a  poorer  conductor  of  the  current,  or,  in 
other  words,  its  electrical  resistance  is  increased.  In  the 


PRIMARY  CELLS  513 

second  place,  the  hydrogen  on  the  surface  of  the  copper 
tends  to  go  into  solution  again  as  positive  ions,  and  conse- 
quently sets  up  an  opposing  E.M.F.  which  retards  the  ap- 
proach of  other  hydrogen  ions  with  their  charges.  This 
action  is  shown  by  the  diminished  P.D.  between  the  plates, 
when  measured  with  a  voltmeter  (an  instrument  to  be 
described  later). 

The  accumulation  of  hydrogen  on  the  positive  plate, 
with  its  attendant  effects,  is  called  the  polarization  of  the 
cell.  In  some  of  the  common  types  of  cells  polarization  is 
diminished  by  the  use  of  chemicals  which  yield  oxygen 
to  the  hydrogen,  forming  water;  in  others  it  is  avoided 
altogether  by  the  use  of  solutions  in  which  the  positive 
charges  are  carried  by  metallic  ions  in  place  of  hydrogen. 

No  single  type  of  cell  is  best  for  all  purposes.  Some  of  the  most 
common  types  are  described  below.  These  descriptions  should  be 
studied  as  opportunity  is  afforded  in  the  class-room  and  the  laboratory 
for  observation  and  use  of  the  different  cells. 

412.  The  Electrical  Resistance  of  a  Cell.  —  All  sub- 
stances, even  the  best  conductors,  offer  greater  or  less  oppo- 
sition to  the  flow  of  electricity  through  them.  The  greater 
this  opposition,  or  electrical  resistance,  the  less  will  be  the 
current  that  a  given  E.M.F.  is  able  to  maintain  in  the  cir- 
cuit. The  effect  of  resistance  in  diminishing  the  current 
is  the  same  whether  it  is  met  with  in  the  external  part  of 
the  circuit  or  within  the  cell.  The  resistance  of  the  plates 
is  negligible;  that  of  the  liquid  may  be  considerable. 

The  resistance  of  the  liquid  varies  with  the  kind  and 
quantity  of  the  materials  in  solution.  With  a  given  liquid 
it  is  reduced  by  using  larger  plates,  and  by  shortening  the 
distance  between  them ;  for  this  provides  a  wider  and  shorter 
path  for  the  current  through  the  liquid. 


514  ELECTRODYNAMICS 

A  cell  supplies  its  greatest  possible  current  through  an 
external  circuit  having  the  least  possible  resistance,  such 
as  a  short  copper  wire.  The  current  is  then  proportional 
to  the  E.M.F.  of  the  cell  and  inversely  proportional  to 
its  resistance.  When  thus  connected  the  cell  is  said  to 
be  short-circuited. 

The  subject  of  resistance  is  considered  in  detail  later. 

413.    The  Chromic  Acid  or  Bichromate  Cell.  — 
The  zinc  plate  of  this  cell  is  attached  to  a  rod,  by 
means  of  which  it  is  raised  from  the  liquid  when  the 
cell  is  not  in  use  (Fig.  383).    The  positive  pole  con- 
sists of  two  plates  of  carbon  —  one  on  each  side  of 
the  zinc  —  which  are  connected  to  the  same  bind- 
ing post  at  the  top.     The  liquid  is  dilute  sulphuric 
acid,  containing  in  solution  chromic  acid  or  bichro- 
mate of  potassium  or  of  sodium,  which  acts  as  a 
depolarizer.    These  substances  contain  oxygen  which 
they  give  up  readily  to  hydrogen,  forming  water, 
chromate  Cell  *~    ^e  accumulation  of  hydrogen  on  the  positive  plates 
is    thus   diminished,   but    not    entirely   prevented. 
Polarization  usually  diminishes  the  current  by  one  third  or  more 
in  a  few  minutes. 

The  electromotive  force  of  this  cell  is  about  two  volts,  which  is 
considerably  higher  than  that  of  most  other  cells.  Its  resistance  is 
small,  for  the  current  has  only  a  very  short  path  in  the  liquid,  and  the 
double  carbon  pole  reduces  the  resistance  further  by  one  half.  Owing 
to  its  high  E.M.F.  and  low  resistance,  the  bichromate  cell  is  capable 
of  supplying  an  exceptionally  strong  current,  and  on  this  account 
is  much  used  in  experimental  work.  The  zinc  should  be  kept  thor- 
oughly amalgamated;  and  it  must  be  raised  from  the  liquid  when  the 
cell  is  not  in  use,  for  the  amalgam  is  only  partially  effective  in  prevent- 
ing local  action. 

414.  The  Leclanche  Cell.  —  The  poles  of  this  cell  are  a  zinc  rod 
and  a  block  of  carbon  (Fig.  384).  The  latter  is  inclosed  in  a  cylin- 
drical cup  of  porous  earthenware,  and  is  packed  round  with  small 


PRIMARY   CELLS 


SIS 


fragments  of  carbon  and  manganese  dioxide  (Mn02).     The  liquid  is 
a  solution  of  ammonium  chloride,  or  sal  ammoniac  (NEUCl),  which 


FIG.  384.  —  Leclanche  Cell. 


FIG.  385.  — Dry  Cell. 


dissociates  into  negative  chlorine  ions,  Cl  ,  and  positive  ammonium 
ions,  NH4+.  When  the  circuit  is  closed,  zinc  ions  displace  the  ammo- 
nium, forming  zinc  chloride  (ZnCk),  which  remains  in  solution.  The 
NH4  ions  move  toward  the  carbon  plate,  where  they  break  up  into 
ammonia  (NHs),  which  dissolves  in  the  solution,  and  hydrogen, 
which  combines  with  part  of  the  oxygen  of  the  MnOo,  forming  water. 
There  is  no  local  action;  hence  the  zinc  is  not  amalgamated.  The 
E.M.F.  of  the  cell  is  about  1.5  volts.  Its  resistance  is  at  least  three  or 
four  times  as  great  as  that  of  the  bichromate  cell,  and  its  maximum 
current  is  correspondingly  less.  In  a  modified  form  of  the  Leclanche 
cell  the  positive  plate  is  made  of  a  mixture  of  carbon  and  man- 
ganese dioxide,  and  the  porous  cup  is  dispensed  with. 

The  action  of  the  manganese  dioxide  is  not  rapid  enough  to  prevent 
the  cell  from  becoming  polarized  if  used  constantly.  Hence  the  cell 
is  satisfactory  only  for  intermittent  service,  as  in  ringing  door  bells. 
It  has  the  merit  of  not  requiring  attention  for  months  at  a  time. 

415.  The  Dry  Cell.  —  In  this  cell  (Fig.  385),  as  in  the  Leclanche 
cell,  the  current  is  due  to  the  chemical  action  of  ammonium  chloride 
on  zinc.  The  zinc  plate  forms  the  containing  vessel,  and  the  solu- 
tion forms  a  porous  paste  with  plaster  of  Paris  and  smaller  quantities 
of  other  materials.  The  E.M.F.  of  the  cell  is  about  1.4  volts.  Its 
resistance  is  very  low;  and,  on  short  circuit,  it  supplies  a  larger  cur- 
rent than  the  bichromate  cell. 


ELECTRODYNAMICS 

The  special  merits  of  the  dry  cell  are  its  convenience  and  its  port- 
ability. It  requires  no  care,  and  when  exhausted  is  thrown  away. 
It  is  extensively  used  on  electric-bell  circuits,  and  supplies  the  current 
for  pocket  electric  lamps  and  for  "spark  ignition"  in  the  gasoline 
engines  of  motor  cycles,  automobiles,  etc. 

416.  The  Gravity  Cell.  —  The  positive  pole  of  this  cell  consists 
of  several  copper  strips  fastened  together,  and  is  placed  at  the  bottom 

of  the  jar  (Fig.  386).  The  zinc  is 
near  the  top,  and  commonly  hangs 
suspended  from  the  edge  of  the  jar. 
The  lower  portion  of  the  liquid  is  a 
saturated  solution  of  copper  sul- 
phate, or  bluestone  (CuSO4).  The 
upper  portion,  in  which  the  zinc  is 
suspended,  is  a  weak  solution  of  zinc 
sulphate.  (Very  dilute  sulphuric  acid 
will  serve  in  setting  up  the  cell.) 
This  is  of  less  specific  gravity  than 
the  lower  solution,  and  rests  upon 
it  without  mixing,  except  by  the 
slow  process  of  diffusion;  hence  the  name  "gravity  cell." 

The  zinc  and  copper  ions  of  the  solution  are  positive;  the  sulphions 
(S04)  are  negative,  as  in  dilute  sulphuric  acid.  When  the  circuit  is 
closed,  copper  ions  pass  out  of  solution  at  the  copper  plate,  upon 
which  they  are  deposited,  giving  up  their  positive  charges.  The  nega- 
tive ions  are  repelled  toward  the  zinc  plate,  where  an  equal  number  of 
positive  zinc  ions  are  passing  into  solution,  leaving  the  zinc  negatively 
charged.  The  solution  of  copper  sulphate  is  continually  renewed 
from  a  supply  of  copper  sulphate  crystals  at  the  bottom  of  the  cell. 
Thus  the  zinc  and  the  copper  sulphate  are  gradually  consumed,  the 
amount  of  zinc  sulphate  in  solution  increases,  and  metallic  copper 
is  added  to  the  copper  plate. 

When  the  cell  is  not  in  use  it  should  be  kept  on  closed  circuit 
through  a  considerable  resistance  (20  to  30  ohms).  The  small  cur- 
rent then  flowing  prevents  diffusion  of  the  liquids  into  each  other; 
otherwise  the  copper  sulphate,  coming  in  contact  with  the  zinc  plate, 
will  deposit  copper  upon  it,  and  the  cell  will  then  furnish  little  or  no 
current.  The  zinc  is  not  amalgamated,  as  there  is  no  local  action. 


FIG.  386.  — Gravity  Cell. 


THE   MAGNETIC  ACTION  OF  A   CURRENT       517 

The  E.M.F.  of  the  gravity  cell  is  about  1.08  volts;  it  is  constant, 
for  the  deposit  of  copper  on  the  copper  plate  does  not  affect  its  elec- 
trical properties  in  any  way.  The  resistance  of  the  cell  is  3  or  4  times 
as  great  as  that  of  the  Leclanche  cell,  and  from  12  to  20  times  that 
of  the  bichromate  cell.  Owing  to  its  low  E.M.F.  and  high  resistance, 
its  maximum  current  is  very  small.  It  is 
especially  serviceable  in  experimental  work 
requiring  a  constant  E.M.F.,  and  is  much 
used  for  purposes  requiring  a  current  all 
or  nearly  all  of  the  time,  as  in  telegraphy. 

417.  The  Daniell  Cell.— The  materials 
in  this  cell  are  the  same  as  in  the  gravity 
cell,  but  they  are  differently  arranged  (Fig. 
387).     The  two  solutions  are  separated  by 
the  walls  of  a  porous  cup,  which  contains 
the  zinc  pole  and  the  zinc  sulphate.     The 
cup  stands  in  the  solution  of  copper  sul- 
phate, and  is  nearly  surrounded  by  a  sheet 

of  copper,  which  serves  as  the  positive  pole.      FIG.  387.  —  Daniell  Cell. 

III.   THE  MAGNETIC  ACTION  or  A  CURRENT 

418.  Oersted's  Experiment.  —  We  have  seen  that  an 
electric  current  affects  a  magnetic  needle,  and  must  there- 
fore be  surrounded  by  a  magnetic  field.     This  great  fact 
was  discovered  by  the  Danish  physicist,  Hans  Christian 
Oersted,  in  1819,  —  nineteen  years  after  Volta's  invention 
of  the  electric  battery.     It  is  related  that,  at  the  close  of 
a  lecture  one  day,  Oersted  held  over  a  magnetic  needle  a 
wire  carrying  a  current,  and  observed,  much  to  his  surprise, 
that  the  needle  set  itself  at  right  angles  to  the  wire  (Fig. 
380).     Thus    was    discovered    the    fundamental    fact    of 
electromagnetism,  which,  in  less  than  a  century,  has  be- 
come one  of  the  controlling  factors  in  the  industries  of  the 
world,  through  the  invention  of  the  telegraph,  the  tele- 
phone, the  dynamo  and  electric  motor,  etc. 


5 1 8  ELECTRODYNAMICS 

Nearly  all  that  follows  in  our  study  of  electricity  has  to 
do  in  one  way  or  another  with  the  magnetic  fields  of  cur- 
rents; and  we  shall  therefore  need  to  become  very  thor- 
oughly acquainted  with  them.  Magnetic  fields  have 
previously  been  considered  in  their  relation  to  the  poles 
of  magnets;  they  are  now  to  be  studied  in  their  relation  to 
the  direction  and  strength  of  electric  currents,  and  the 
shape  of  conductors. 

419.  The  Magnetic  Field  of  a  Current  in  a  Straight 
Conductor.  —  The  magnetic  lines  of  force  about  a  wire 
can  be  shown  with  iron  filings,  provided  the  current  is 
quite  strong.  A  battery  of  three  or  four  bichromate  cells, 
connected  in  parallel  (Art.  448),  will  generate  a  sufficient 
current;  but  a  single  cell  is  as  good  or  better  when  the  con- 
necting wire  is  formed  into  a  large  coil  of  some  15  or  20 
turns,  placed  close  together.  With  a  given  current  the 
field  is  strengthened  in  proportion  to  the  number  of  parallel 
wires,  but  in  other  respects  it  is  the  same  as  if  there  were 
only  a  single  wire.  Following  either  plan,  let  a  current 
be  sent  through  a  straight,  vertical  conductor,  which  pierces 
a  sheet  of  cardboard.  Iron  filings  sprinkled  on  the  card- 
board show  that  the  lines  of  force  are  concentric  circles 
about  the  wire  (Fig.  388).  The  lines  of  filings  are  most 
distinct  close  to  the  wire,  and  at  a  distance  of  a  few  inches 
none  are  formed.  Evidently  the  field  grows  weaker 
from  the  wire  outward  in  all  directions;  but  it  has  the  same 
strength  at  a  given  distance  on  all  sides  and  all  along  the 
wire.  The  magnetic  field  is  in  the  form  of  a  cylinder,  with 
the  wire  extending  along  its  axis;  and  each  circular  line  of 
force  lies  in  a  plane  at  right  angles  to  the  wire. 

The  direction  of  the  lines  of  force  round  the  wire  depends 
on  the  direction  of  the  current.  If  the  current  is  flowing 


THE  MAGNETIC  ACTION  OF  A   CURRENT       519 

upward,  the  north  pole  of  a  magnetic  needle  points  counter- 
clockwise round  the  wire,  as  shown  in  the  figure;  if  it  is 
flowing  downward,  the  north  pole  of  the  needle  points 


FIG.  388.  —  Direction    of   Lines   of 

Force  about  a  Wire  in  which  FIG.   389.  —  The  Right- 

the  Current  is  flowing  upward.  hand  Rule. 

clockwise  round.  There  is,  then,  a  definite  and  invariable 
relation  between  the  direction  of  the  current  and  the  direc- 
tion of  the  magnetic  lines  of  force.  This  relation  is  most 
serviceably  stated  in  terms  of  the  right-hand  rule:  Grasp 
the  wire  with  the  right  hand,  with  the  extended  thumb 
pointing  in  the  direction  of  the  current;  then  the  fingers 
will  point  round  the  wire  in  the  direction  of  the  lines  of 
force  (Fig.  389). 

With  the  aid  of  this  rule  we  can  determine  the  direction 
of  the  current  in  a  wire  by  observing  the  deflection  of  a 
compass  needle  near  it;  and,  conversely,  if  we  know  the 
direction  of  the  current,  the  rule  gives  the  direction  of  the 
lines  of  force. 

420.  The  Magnetic  Field  of  a  Current  in  a  Circular 
Coil.  —  The  magnetic  lines  of  force  about  a  curved  con- 
ductor are  crowded  together  on  the  concave  side  and  spread 
apart  on  the  convex  side  (Fig.  390).  Their  direction  round 


520 


ELECTRODYNAMICS 


any  part  of  the  conductor  bears  the  same  relation  to  the 
direction  of  the  current  as  in  a  straight  wire.  All  the  lines 
within  a  loop  extend  through  it  in  the  same  direction,  as 
shown  in  the  figure. 

The  field  at  the  center  of  a  coil  is  especially  important, 
since  this  is  where  the  magnetic  needle  is  placed  in  instru- 
ments for  measuring  the  strength  and  the 
voltage  of  electric  currents.  When  a  strong 
current  is  sent  through  a  coil  of  fifteen  or 
more  turns,  the  lines  of  force  within  and 
about  the  coil  can  be  shown  with  iron 
filings  (Fig.  391).  At  the  center  the  lines 
are  straight,  and  perpendicular  to  the  plane 
of  the  coil.  Their  direction  relative  to 
the  direction  of  the  current  is  given  by 
the  right-hand  rule  for  coils:  Close  the 
right  hand  and  place  it  within  the  coil, 
with  the  fingers  pointing  round  in  the  direction  of  the  cur- 
rent; then  the  extended  thumb  will  point  in  the  direction 
of  the  lines  of  force  through  the  coil. 


FIG.  390.  —  Mag- 
netic  Field 
about  a  Coil. 


FIG.  391.  — Section  of  the  Field  of  a 
Coil. 


FIG.    392.  —  Magnetic    Field    of    a 
Helix. 


421.  The  Helix  and  the  Electromagnet.  —  An  elongated 
cylindrical  coil  of  wire  is  called  a  helix  or  solenoid.  If  it 
is  made  of  bare  wire,  it  must  be  wound  with  an  open  space 
between  adjacent  turns,  in  order  to  carry  an  electric  cur- 


THE  MAGNETIC  ACTION  OF  A   CURRENT        521 

rent  properly  (Fig.  392).  With  insulated  wire,  it  may  be 
close-wound  in  one  or  more  layers  of  turns;  and  the  insu- 
lation compels  the  current  to  travel  round  each  turn  in 
succession,  from  one  end  of  the  wire  to  the  other.  When 
such  a  coil  is  wound  round  a  bar  or  rod  of  soft  iron,  the 
iron  is  called  a  core,  and  the  core  and  coil  together  form  an 
electromagnet.  We  shall  see  that  electromagnets  play  the 
leading  role  in  the  generation  and  use  of  electric  currents 
in  daily  life;  without  them  electricity  would  be  of  very 
little  use  indeed. 

The  action  of  an  electromagnet  depends,  in  the  first 
place,  on  the  magnetic  field  of  the  helix.  This  field  is 
similar  to  that  of  a  flat  coil,  as  described  above,  the  only 
difference  being  that  it  is  elongated  in  the  direction  of  the 
axis.  Within  the  coil  the  lines  of  force  extend  from  end 
to  end  in  approximately  straight  lines.  Outside  the  coil 
the  field  is  like  that  of  a  bar  magnet.  The  lines  spread  out 
from  one  end  and  return  to  the  other,  each  line  forming 
a  closed  curve.  The  helix,  when  a  current  is  flowing  in 
it,  behaves  like  a  magnet.  The  end  from  which  the  lines 
of  force  emerge  repels  the  north  pole  of  a  needle  and 
attracts  the  south  pole.  This  is  the  north  end  or  pole  of 
the  helix.  The  other  end  is  a  south  pole.  If  a  helix  is 
supported  so  that  it  is  free  to  turn  in  a  horizontal  plane, 
while  carrying  a  current,  it  turns  into  a  north-and-south 
line,  like  a  compass  needle.  The  direction  of  the  current 
round  a  helix  determines  its  polarity,  in  agreement  with 
the  right-hand  rule  for  coils:  If  the  helix  is  grasped  in 
the  right  hand  so  that  the  fingers  point  round  it  in  the  direc- 
tion of  the  current,  the  extended  thumb  will  point  toward 
the  north  pole  of  the  helix  (Fig.  393).  It  should  be  noted 
that,  if  two  coils  are  oppositely  wound,  the  current  enters 
at  the  north  end  of  one"  and  at  the  south  end  of  the  other. 


522  ELECTRODYNAMICS 

The  magnetic  effects  of  a  helix  are  greatly  intensified 
when  a  soft  iron  core  is  inserted.  The  magnetic  field  of 
the  coil  induces  magnetism  in  the  core,  with  like  poles  of 

the  core  and  coil  at  the  same 
end.  If  the  strength  of  the  cur- 
rent is  sufficient,  the  core  will 
be  magnetized  to  saturation, 
and  its  magnetic  strength  will 

FIG.  393.  — Right-hand  Rule.       ,          r 

be    from    1000  to    2000    times 

as  great  as  that  of  the  helix  alone,  and  many  times  as 
great  as  that  of  a  permanent  steel  magnet  of  the  same 
size.  By  testing  the  power  of  the  core  to  hold  iron 
filings  or  nails,  it  will  be  found  that  the  coil  is  instantly 
magnetized  when  the  current  is  started,  and  instantly 
demagnetized  when  the  current  is  stopped.  An  electro- 
magnet is  thus  under  the  perfect  control  of  the  current. 
It  is  this  property  of  controllability  which  gives  to  electro- 
magnets their  wide  field  of  usefulness.  The  property  next 
in  importance  is  their  great  strength. 

Both  of  these  properties  are  strikingly  shown  by  the  large  lifting 
magnets,  which  are  now  widely  used  in  iron-foundries  and  machine- 
shops  for  lifting  iron  and  steel  castings,  etc.  In  Fig.  394  is  shown  one 
of  the  smallest  of  these  magnets  in  the  act  of  lifting  an  8oo-lb.  load. 
The  magnet  itself  (the  low  cylinder  just  below  the  hook)  is  10  in. 
in  diameter  and  weighs  75  Ib.  Magnets  from  50  to  60  in.  in  diam- 
eter and  capable  of  lifting  from  20,000  to  50,000  Ib.  are  now  in  daily 
use.  A  lifting  magnet  is  suspended  from  a  crane  by  means  of  chains 
and  pulleys,  so  that  it  can  be  raised  and  lowered  and  moved  from 
place  to  place.  The  closing  or  opening  of  a  switch,  turning  the  elec- 
tricity on  or  off,  causes  the  magnet  to  pick  up  or  release  its  load. 

422.  The  Strength  of  Electromagnets.  —  For  an  electromagnet 
to  attain  its  full  strength,  the  current  sent  through  the  coil  must  have 
a  certain  strength.  A  weaker  current  will  only  partly  magnetize 
the  core.  On  the  other  hand,  when  the  core  is  already  magnetized 
to  saturation,  a  further  increase  of  current  has  practically  no  effect. 


THE  MAGNETIC  ACTION  OF  A  CURRENT       523 

The  greater  the  number  of  turns  in  the  coil  the  smaller  will  be 
the  current  required  to  produce  an  equal  magnetization  of  the  core. 
For  the  core  is  magnetized  by  the  mag- 
netic field  of  the  current,  and  the  inten- 
sity of  this  field  is  proportional  jointly 
to  the  strength  of  the  current  and  the 
number  of  turns.  Hence  if  the  number 
of  turns  is  increased,  the  strength  of 
the  current  may  be  decreased,  and  vice 
versa.  For  certain  purposes  the  coils 
of  electromagnets  are  wound  with 
hundreds  or  thousands  of  turns  of  fine 
wire.  These  require  only  a  very  small 
current.  For  other  uses  the  coil  has 
only  a  few  turns,  and  a  proportionately 
larger  current  is  necessary. 

The  strength  of  an  electromagnet 
depends  also  upon  the  quality  of  iron 
used  in  the  core.  The  softest  iron  can 
be  most  strongly  magnetized  and  re- 
quires the  least  current.  It  is  also  the 
most  completely  demagnetized  when  the  current  is  turned  off. 

The  shape  of  an  electromagnet  affects  its  strength,  as  well  as  its 
adaptability  to  particular  uses.     Other  conditions  being   the  same, 

the  shorter  the  air  gap  be- 
tween the  poles  the  greater 
will  be  the  strength;  hence  the 
horseshoe  form  is  generally 
preferable  (Fig.  395).  When 
the  two  poles  act  together  on 
the  same  mass  of  iron,  A ,  the 
attraction  is  much  more  than 
twice  that  of  either  pole  alone; 
for  the  iron  virtually  brings 
the  poles  of  the  magnet  together,  and  each  strengthens  the  other 
by  induction.  The  coils  on  the  two  arms  of  such  a  magnet  are 
oppositely  wound,  as  shown  in  the  figure.  (Why?) 

A  bar  of  soft  iron  extending  across  between  the  poles  of  a  magnet, 
either  in  contact  with  them  or  near  them,  is  called  an  armature. 


FIG.  394.  —  Ten-inch  Magnet 
Lifting  800  Pounds. 


FIG.  395.  —  Horseshoe  Electromagnet. 


524 


ELECTRODYNAMICS 


423.  The  Electric  Bell  (Fig.  396)  is  a  simple  and  familiar 
application  of  the  electromagnet.  Bells  differ  more  or 
less  in  details  of  construction,  but  the  essential  parts  are 
the  same  in  all.  Generally,  as  a  matter  of  convenience  in 
construction,  the  electric  circuit  includes  a  short  path 
through  the  metal  frame  of  the  bell.  In  any  case  the  con- 
nections and  insulations  must  be  such  that  the  only  path 
offered  the  current  through  the  bell  is  by  way  of  the  coils 
of  the  electromagnet  and  across  between  the  free  end  of  a 
spring^  a,  and  the  end  of  a  screw,  c. 
The  spring  s,  which  carries  the  arma- 
ture, is  so  adjusted  that  it  holds  the 
armature  away  from  the  magnet,  and, 
at  the  same  time,  presses  the  spring  a 
against  the  screw.  When  the  circuit  is 
closed  by  pressing  a  push  button,  placed 
at  some  convenient  point  in  the  circuit, 
the  electromagnet  attracts  the  arma- 
ture, and  the  clapper  attached  to  it 
strikes  the  bell.  At  the  same  time  the 
spring  is  pulled  away  from  the  screw 
at  c,  thus  breaking  the  circuit.  With 
the  stopping  of  the  current,  the  magnet  releases  the  arma- 
ture, which  is  then  pulled  back  by  the  spring  s.  This 
closes  the  circuit  at  c  again,  and  the  process  is  repeated  as 
long  as  the  push  button  is  pressed. 

A  simplified  diagram  of  the  bell  and  the  electric  circuit 
is  shown  in  Fig.  397.  In  this  figure  the  binding  post  B  is 
connected  with  the  armature,  while  in  the  preceding  figure 
it  is  connected  with  the  screw.  It  is,  of  course,  immaterial 
whether  the  current  passes  from  the  armature  to  the  screw 
or  vice  versa.  In  either  case'  the  armature  acts  as  an 
automatic  circuit-breaker,  which  is  the  essential  thing. 


FIG.  396.  —  Electric  Bell. 


THE   MAGNETIC  ACTION  OF  A   CURRENT       525 

424.  The  Electromagnetic  Telegraph  was  the  first  great 
industrial  triumph  of  electricity.  Its  invention  followed 
close  upon  the  experimental  researches  of  Joseph  Henry, 
of  Albany,  New  York,  on  the  electromagnet.  The  earli- 
est electromagnets  were 
wound  with  a  single 
layer  of  bare  copper 
wire,  insulated  wire 
being  then  unknown. 
Henry  covered  his 
magnet  wire  with  silk, 
and  constructed,  coils 
of  many  layers  of  turns 
(1830).  He  then  began 
to  experiment  on  the 
transmission  of  signals 
by  the  action  of  an 
electromagnet  at  a  dis- 
tance; but  he  soon  be- 
came engrossed  in  other 
lines  of  investigation, 
and  it  was  left  for 
others  to  work  out  the  practical  application  of  his  impor- 
tant discoveries.  And  others  there  were,  both  in  England 
and  America,  who  took  up  the  problem  with  more  or  less 
successful  results.  In  the  United  States  the  successful 
inventor  was  Samuel  F.  B.  Morse,  of  New  York,  who 
constructed  his  first  practical  instruments  in  1838.  The 
first  commercial  telegraph  line  in  the  United  States  was 
built  by  Morse  between  Washington  and  Baltimore  in 
1844. 

At  the  receiving  station,  in  the  original  Morse  system, 
the  message  was  automatically  recorded  in  dots  and  dashes 


Clapper 


Gong 
FIG.  397.  —  Diagram  of  a  Bell  Circuit. 


526  ELECTRODYNAMICS 

on  a  strip  of  paper;  but  operators  soon  found  that  they 
could  read  messages  by  listening  to  the  clicking  sounds 
of  the  recording  instrument.  This  method  was  soon 
adopted,  and  the  receiving  instrument  was  modified  into 
a  sounder. 

425.  The  Sounder  (Fig.  398)  is  operated  by  an  electro- 
magnet, the  poles  of  which  point  upward.     An  armature 
of  soft  iron  is  fixed  across  a  lever  just  above  the  poles. 
When  a  current  passes  through  the  coils  of  the  magnet, 

the  armature  is  attracted 
down,  carrying  the  lever 
with  it,  and  a  screw 
near  the  end  of  the 
lever  makes  a  click  as  it 
strikes  the  support  be- 
neath. When  the  cir- 
cuit is  broken  the 
armature  is  released. 

FIG.  398.  —  Telegraph  Sounder.  .  , 

and  a  spring  throws  the 

lever  back  against  a  screw  above  it.  The  two  clicks  of 
the  lever  sound  differently  and  are  thus  easily  dis- 
tinguished from  each  other.  They  together  constitute  a 
"dot"  when  one  follows  immediately  after  the  other,  and 
a  "dash"  when  there  is  a  brief  interval  between  them. 
The  letters  of  the  alphabet,  the  punctuation  marks,  and 
the  numbers  from  zero  to  nine  are  represented  by  certain 
combinations  of  dots  or  dashes,  or  of  dots  and  dashes 
together. 

426.  The  Key  (Fig.  399)  is  a  device  by  which  the  oper- 
ator makes  and  breaks  the  circuit  in  the  act  of  sending  a 
message.     It  is  fastened  to  a  table  by  two  screws,  the  one 
at  the  left  in  the  figure  being  insulated  from  the  metal  base. 


FIG.  399.  —  Telegraph  Key. 


THE   MAGNETIC  ACTION  OF  A  CURRENT       527 

One  wire  of  the  line  is  fastened  to  each  screw.  There  is 
a  small  platinum  point  at  the  top  of  the  insulated  sciew, 
and  another,  P,  just  above  it  on  the  under  side  of  the  lever. 
The  circuit  is  closed  by 
the  contact  of  these 
points,  when  the  lever 
is  depressed.  When  the 
key  is  not  in  use,  the 
circuit  is  kept  closed  by 
the  switch  5,  which  con- 
nects the  base  of  the 
instrument  with  the  insulated  post,  as  shown  in  the  figure. 
This  switch  is  moved  to  the  right  while  a  message  is  being 
sent,  leaving  the  circuit  open  and  under  the  control  of  the 
operator  by  means  of  the  Jever.  On  a  short  line  a  key 
and  a  sounder  at  each  station  are  the  only  instruments 
required,  and  all  are  in  the  same  circuit. 

427.  The  Relay.  —  Owing  to  the  high  resistance  of  a  long  tele- 
graph line,  the  current  is  too  weak  to  operate  a  sounder.  This  diffi- 
culty might  be  overcome  by 
using  a  battery  of  a  very 
great  number  of  cells;  but  it 
is  more  convenient  and  more 
economical  to  make  use  of 
an  additional  instrument, 
called  a  relay  (Fig.  400). 
The  relay  acts  on  the  same 
principle  as  the  sounder,  but 

the  electromagnet  is  horizontal  and  the  armature  lever  vertical. 
The  lever  is  light  and  delicately  balanced,  and  responds  to  much 
smaller  forces  than  the  lever  of  the  sounder  does. 

The  coils  of  the  magnet  are  connected  with  the  line  circuit,  which 
runs  to  the  distant  station.  This  connection  is  made  at  two  binding 
posts,  A  and  B,  at  which  the  ends  of  the  magnet  wire  terminate. 
A  second  pair  of  binding  posts,  C  and  D,  connects  the  relay  with 


FIG.  400.  —  Telegraph  Relay. 


528 


ELECTRODYNAMICS 


a  local  circuit,  containing  the  sounder  and  a  battery  to  operate  it. 
This  circuit  runs  from  one  post  to  the  armature  lever,  and  from  the 
other  post  to  the  screw  which  the  upper  end  of  the  lever  strikes 
when  drawn  over  by  the  electromagnet.  This  contact  closes  the 
local  circuit.  When  the  line  circuit  is  broken,  the  lever  is  pulled 
back  by  a  spring,  and  strikes  an  insulating  stop  at  the  end  of  the 
opposite  screw.  The  local  circuit  is  then  open.  Thus  when  the 
line  circuit  is  closed  or  opened  by  means  of  the  key,  the  local 
circuit  is  at  the  same  instant  closed  or  opened  by  the  action  of  the 
relay;  and  the  message  is  read  from  the  sounder.  The  relay  alone 
would  not  serve,  as  its  sounds  are  scarcely  audible. 

428.   A  Complete  Telegraph  Line.  —  A  diagram  of  a  complete 
telegraph  line  connecting  two  cities  is  shown  in  Fig.  401.     In  actual 


New  York 


Sounder 


Philadelphia 
Sounder  Re 


Local  Battery 


Local  Battery 


Earth  Earth 

FIG.  "401.  —  Diagram  of  Telegraph  Circuit. 

practice  there  are  generally  many  stations  on  the  same  circuit,  and 
at  each  station  the  line  wire  connects  with  a  key  and  the  magnet 
coils  of  a  relay.  At  the  terminal  stations  the  line  wire  is  connected, 
with  the  earth  by  means  of  metal  plates  sunk  in  moist  ground.  The 
earth  completes  the  circuit,  taking  the  place  of  a  return  wire.  There 
is  a  line  battery  *  at  each  terminal  station,  consisting  of  many  cells 
in  series  (Art.  448).  Since  the  circuit  remains  closed  when  the  line 
is  idle,  a  non-polarizing  cell  is  required,  such  as  the  gravity  cell. 
Small  dynamos  are  now  very  generally  used  on  long  lines  instead 
of  batteries.  A  local  battery  supplies  the  current  for  the  sounder 
at  each  station. 

*  In  diagrams  of  electric  circuits  a  cell  is  commonly  represented  by  the 
symbol  1 1 ,  the  long  thin  line  representing  the  positive  plate  and  the  short 
thick  line  the  negative  plate.  A  battery  is  represented  by  a  series  of  these 
symbols,  one  for  each  cell. 


THE  MAGNETIC  ACTION  OF  A  CURRENT        529 

When  an  operator  wishes  to  send  a  message,  he  opens  the  switch 
of  his  key  and  calls  the  receiving  station.  The  sounders  at  all  the 
stations  deliver  the  message,  but  the  operator  at  the  station  called 
is  the  only  one  who  pays  attention  to  it. 

429.  Later  Developments  in  Telegraphy.  —  By  the  system  of 
telegraphy  described  above,  only  one  message  at  a  time  can  be 
sent  over  a  wire.  More  complicated  instruments  are  now  in  general 
use  by  means  of  which  four  messages  can  be  sent  at  one  time,  two 
in  each  direction,  over  the  same  wire.  This  is  known  as  quadruplex 
telegraphy.  Eight  operators  are  employed  on  each  wire,  one  to  send 
and  one  to  receive  each  of  the  four  messages.  A  skilled  operator  can 
signal  about  35  words  per  minute;  hence  by  the  quadruplex  system 
the  capacity  of  a  single  wire  is  about  140  words  per  minute. 

Even  at  this  rate  many  wires  are  required  for  the  ordinary  business 
of  telegraph  offices  in  large  cities;  and  other  lines  are  occupied  in 
transmitting  long  press  despatches  for  the  daily  papers.  These  great 
and  growing  demands  have  led  to  the  invention  of  various  systems  of 
high-speed  telegraphy,  in  which  both  the  sending  and  the  receiving 
instruments  operate  automatically.  In  the  Barclay  printing-tele- 
graph system  the  messages  are  first  punched  in  the  Morse  charac- 
ters on  a  long  tape,  by  a  special  form  of  typewriter.  The  tape  is 
then  fed  into  the  sending  machine.  The  holes  in  the  tape  allow  elec- 
trical contact  to  be  made,  which  sends  impulses  over  the  wire  just 
as  they  are  sent  by  an  ordinary  operator's  key,  only  much  faster. 
One  wire  will  transmit  messages  as  fast  as  three  or  four  girls  can  perfo- 
rate the  tapes.  At  the  receiving  end  of  the  wire,  an  electrically  oper- 
ated typewriter  takes  the  message  and  prints  it  in  letters  instead  of 
dots  and  dashes.  This  is  done  automatically,  and  no  operator  is 
required  at  the  receiving  end. 

Another  recent  system,  invented  by  two  Hungarian  electricians, 
Anton  Pollak  and  Josef  Virag,  also  makes  use  of  a  perforated  strip 
in  the  sending  instrument.  The  receiver  records  the  message  on 
photographic  paper,  by  means  of  a  pencil  of  light  reflected  from  a 
tiny  mirror.  This  weightless  pencil  writes  the  message  in  ordinary 
script  at  the  rate  of  800  words  per  minute,  or  four  times  as  fast  as 
a  person  can  talk! 

The  Delany  telepost  system,  now  in  operation  in  New  England  and 
the  middle  West,  is  still  more  rapid,  its  "ordinary"  rate  being  1,000 


530 


ELECTRODYNAMICS 


words  per  minute  and  its  maximum  2000  or  more.  The  sending 
instrument  is  operated  automatically  by  means  of  a  perforated 
tape,  as  in  the  other  automatic  systems.  At  the  receiving  end  the 
message  is  recorded  in  dots  and  dashes  on  a  chemically  prepared 
tape. 

430.    Open-circuit    Systems   for   Amateurs.  —  Boys  who  would 
like  to  set  up  a  telegraph  line  for  their  own  amusement  or  instruc- 


FIG.  402.  —  Diagram  of  an  Open-circuit  Telegraph  Line. 

tion  will  find  it  less  expensive  to  adopt  either  of  the  open-circuit 
systems  shown  in  Figs.  402  and  403.  In  either  case  the  battery  is 
in  action  only  while  a  message  is  being  sent,  and  dry  cells  may  be  used. 
By  the  method  shown  in  Fig.  402  the  batteries  at  the  two  stations 
are  connected  so  as  to  oppose  each  other.  The  keys  are  left  open 

.    .  *3oi/acJer 


Line* 


FIG.  403.  —  Diagram  of  an  Open-circuit  Telegraph  Line. 

when  not  in  use;  and  the  circuit  is  then  closed  through  the  two  bat- 
teries, the  sounders,  the  line  wire,  and  the  ground.  There  is  no  cur- 
rent, since  the  batteries  are  opposed.  Closing  the  key  at  either 
station  operates  both  sounders.  (Explain.) 

The  system  shown  in  Fig.  403  requires  a  two-point  switch  at  each 
station  in  addition  to  the  key  and  sounder.  When  the  line  is  not  in 
use,  the  switches  are  turned  so  as  to  close  the  circuit  through  the 


MEASUREMENT  OF  ELECTRIC   CURRENTS       531 

sounder  at  each  station,  but  not  through  the  batteries;  hence  there 
is  no  current.  The  person  who  wishes  to  call  throws  his  switch  so 
that  it  connects  the  line  with  the  battery  and  key  at  his  station. 
He  can  then  call  the  other  station  and  send  messages  in  the  usual 
manner.  To  answer  the  call,  the  other  person  must  throw  the  switch 
at  his  station  so  as  to  include  his  own  battery  and  key  in  the  circuit. 
(Explain  in  detail.) 

IV.  MEASUREMENT  OF  ELECTRIC  CURRENTS 

431.  Current  Strength.  —  By  the  strength  of  an  electric 
current  is  meant  the  quantity  of  electricity  which  flows 
past  any  point  of  the  circuit  in  one  second.  This  quantity 
is  the  same  at  all  points  along  an  .undivided  circuit,  regard- 
less of  the  kind  or  amount  of  work  which  the  current  may 
be  doing.  Thus  if  a  small  elec- 
tric lamp  is  lighted  by  the  cur- 
rent from  a  battery  (Fig.  404), 
electrical  energy  is  lost  from  the 
circuit  by  transformation  into 
heat  in  the  filament  of  the 
lamp;  but  there  is  just  as 
much  current  after  it  has  passed 
through  the  lamp  as  before  (see 

Art.  400).     This  is  proved  by 

•       *u  vu  FlG-  4°4- 

measuring  the  current  with  an 

ammeter  (Art.  435)  before  it  reaches  the  lamp  and  again 
after  it  has  passed  through.  It  may  also  be  shown  in 
a  very  simple  way  by  lighting  two  or  more  small  lamps 
in  series  (i.e.  placed  one  after  the  other  in  the  circuit),  all  the 
lamps  being  alike  (Fig.  405).  The  number  of  cells  required 
will  be  in  proportion  to  the  number  of  lamps;  but  the  sig- 
nificant fact  for  our  present  purpose  is  that  all  the  lamps 
are  equally  lighted,  the  last  of  the  series  as  brilliantly  as 
the  first.  This  proves  (as  nearly  as  the  eye  can  judge) 


532 


ELECTRODYNAMICS 


FIG.  405. 


that  the  current  continues  undiminished  through  all  the 
lamps. 

The  practical  unit  of  electric  current  is  called  the  ampere, 
after  the  French  physicist,  Andre  Marie  Ampere  (1775- 
1836),  who  made  important  discoveries  concerning  the 

magnetic  action  of  cur- 
rents. The  ampere  is  de- 
fined as  the  current  that 
would  produce  a  certain 
magnetic  or  a  certain  chem- 
ical effect.  These  defini- 
tions are  of  the  greatest 
scientific  and  practical  im- 
portance; but  they  are  of  no 
service  to  the  beginner  in  electrical  science,  since  they  relate 
to  matters  with  which  he  is  not  familiar.  A  definite  or 
even  approximate  idea  of  the  ampere  and  other  electrical 
units  can  be  gained  only  through  personal  acquaintance 
with  the  effects  of  currents  under  known  conditions.  This 
acquaintance  will  come,  in  some  measure,  through  the 
experiments  of  the  class-room  and  the  laboratory. 

Compared  with  the  quantities  of  electricity  present  in  electro- 
static phenomena,  the  quantity  carried  in  one  second  by  a  current  of 
one  ampere  is  enormous  (Art.  405).  For  example,  a  flow  of  one  am- 
pere for  one  two-hundred-thousandth  of  a  second  would  be  sufficient 
to  charge  an  insulated  sphere  a  foot  in  diameter  to  a  potential  of 
300,000  volts.  Yet  a  single  dry  cell,  on  short  circuit,  will  supply  a 
current  of  15  amperes  or  more. 

Very  little  can  be  gained,  however,  by  comparing  plectric  cur- 
rents with  electric  charges.  The  most  serviceable  ideas  of  current 
strength  are  derived  from  a  knowledge  of  the  effects  which  currents 
produce;  but  it  should  be  noted  that  the  voltage  of  a  current  is  also 
a  determining  factor  in  producing  these  effects.  In  short,  the  power 
of  a  current  is  proportional  jointly  to  its  amperage  (strength)  and  its 
voltage  (pressure),  just  as  water-power  is  proportional  jointly  to  the 


MEASUREMENT  OF   ELECTRIC   CURRENTS       533 

quantity  of  the  flow  per  second  and  the  pressure  or  head.  The  power 
of  a  current  of  one  ampere,  maintained  by  an  E.M.F.  of  one  volt, 

is  equal  to  — r  horse-power,  or  .74  ft.-lb.  per  second;  but  the  same 
740 

current,  when  maintained  by  an  E.M.F.  of  746  volts,  transmits  one 
horse-power.  (These  matters  will  receive  further  attention  later.) 

432.  Methods  of  Measuring  Currents. — Electric  currents 
can  be  measured  by  means  of  their  heating,  chemical,  or 
magnetic  effects  (Art.  402);  but  the  magnetic  effect  is  the 
only  one  adapted  to  general  use.     Instruments  which  meas- 
ure currents  by  their  magnetic  effects  are  called  galvanome- 
ters.  These  are  of  various  forms;  but  their  action  in  all  cases 
depends  upon  the  fact  that  the  magnetic  field  of  a  current  in 
a  given  circuit  is  proportional  to  the  strength  of  the  current. 

433.  The    Tangent    Galvanometer.  —  In    the  tangent 
galvanometer  (Fig.  406)  the  current  to  be  measured  is  sent 
through  a   vertical,  circular   coil    of 

insulated  wire.  A  short,  magnetic 
needle  is  mounted  with  its  center  at 
the  center  of  the  coil.  A  long,  non- 
magnetic pointer  is  attached  at  right 
angles  to  the  needle  and  turns  with 
it  (Fig.  407).  Any  deflection  of 
the  needle  is  indicated  by  the 
pointer  as  it  moves  over  a  dial  grad- 
uated in  degrees.  FIG.  406.— Tangent  Gal- 
T  .  vanometer. 

In  using  a  tangent  galvanometer, 

it  must  be  turned  so  that  the  plane  of  the  coil  is  in  the  mag- 
netic north-and-south  line.  This  adjustment  is  made  while 
there  is  no  current  in  the  coil.  The  coil  and  the  needle  are 
then  in  the  same  vertical  plane,  and  the  ends  of  the  pointer 
are  at  the  zero  points  of  the  scale  on  the  dial. 

When  the  galvanometer  is  connected  in  a  circuit,  the 


534 


ELECTRODYNAMICS 


FIG.    407.  —  Compass 
of  Galvanometer. 


magnetic  field  of  the  current  in  the  coil  acts  on  the  needle 
and  tends  to  turn  it  at  right  angles  to  the  plane  of  the  coil 
(Fig.  391).  But  the  deflection  of  the 
needle  is  opposed  by  the  earth's  field. 
Each  pole  of  the  needle  is  thus  acted 
upon  by  two  forces  at  right  angles  to  each 
other  (Fig.  408),  and  the  needle  comes  to 
rest  in  line  with  the  resultant  of  these 
forces.  Since  the  force  due  to  the  cur- 
rent is  proportional  to  the  current,  it  is 
evident  that  the  angle  through  which  the  needle  turns 
(called  the  deflection)  will  be  greater  or  less  according  as 
the  current  is  stronger  or  weaker.  The  current,  however, 
is  not  proportional  to  the  angle  of  deflection,  but  to  the 
tangent  of  the  angle ;  hence  the  name  tangent  galvanometer. 
(See  Lab.  Ex.  63  for 
a  discussion  of  this 
relation,  and  detailed 
directions  for  the  use 
of  the  instrument.) 

The  value  of  a  cur- 
rent in  amperes  can  be 
obtained  by  multiplying 
the  tangent  of  the  angle 
of  deflection  by  a  factor, 
which  is  a  constant  for  the  same  instrument  at  the  same  place,  or 
the  scale  can  be  graduated  to  read  in  amperes  directly.  This 
graduation,  if  correct  for  one  locality,  would  be  incorrect  for  another, 
unless  the  earth's  magnetic  field  happened  to  be  of  equal  intensity 
at  the  two  places.  (Why?) 

434.  The  D'Arsonval  Galvanometer.  —  There  are  two 
principal  types  of  galvanometers.  In  the  one  the  current 
is  sent  through  a  fixed  coil,  and  a  magnetic  needle  is  de- 


FIG.  408.  —  Component   and   Resultant  Forces  on 
Galvanometer  Needle. 


N 


MEASUREMENT  OF  ELECTRIC  CURRENTS       535 

fleeted.  The  tangent  galvanometer  is  an  example.  In  the 
other  type  the  magnet  is  fixed  and  the  coil  is  deflected.  Such 
instruments  are  called  D' Arson val  galva- 
nometers, after  the  French  scientist  who 
originated  this  type.  They  are  made  in 
a  great  variety  of  forms,  adapted  to  use 
under  different  conditions. 

The  general  plan  of  a  laboratory  or 
a  lecture-table  D' Arson  val  is  shown  in 
Fig.  409  and  a  complete  lecture-table 
instrument  in  Fig.  410.  A  coil  of  fine 
wire  hangs  between  the  poles  N  and  S 
of  a  strong  permanent  magnet.  The 
circuit  includes  the  coil,  the  slender  metal 
ribbon  by  which  it  is  suspended,  and  a 
similar  ribbon,  in  the  form  of  a  spiral, 

'  .  r        .         ,.         FlG-  409-  —  Diagram 

leading  from  it  beneath.  A  fixed  cyhn-  ofD'ArsonvaiGai- 
drical  core  of  soft  iron  is  mounted  within  variometer. 
the  coil  to  strengthen  the  magnetic  field.  When  no  current 
is  passing,  the  connecting  ribbons  hold  the  coil  so  that  its 
plane  is  parallel  to  the  line  joining  the  poles  of  the  magnet; 
but,  with  a  current  flowing,  the  coil  tends  to  set  itself  with 
its  north  side  facing  the  south  pole  of  the  magnet.  This 
rotation  is  opposed  by  the  torsion  of  the  ribbons,  and  the 
coil  turns  through  a  greater  or  less  angle,  depending  upon 
the  strength  of  the  current.  In  some  instruments  the  coil 
carries  a  non-magnetic  pointer,  which  moves  over  a  scale 
(Fig.  410);  in  others  it  carries  a  small  mirror,  M  (Fig.  409), 
which  indicates  the  deflection  by  the  angle  at  which  it 
reflects  a  beam  of  light  or  the  image  of  a  scale  placed  at  some 
distance  in  front  of  it. 


One  important  advantage  of   the  D'Arsonval  galvanometer  is 
that   it   is   independent   of   the   earth's  field,    which  is   negligible 


536 


ELECTRODYNAMICS 


in  comparison  with  the  strong  field  of  the  magnet;  hence  the 
instrument  does  not  need  to  be  turned  in  any  particular  direc- 
tion. The  sensitiveness  of  this 
galvanometer  is  increased  by  de- 
creasing the  size  of  the  supporting 
ribbon,  by  increasing  the  strength 
of  the  magnet,  or  by  increasing 
the  number  of  turns  of  the  coil. 
With  the  most  sensitive  instru- 
ments a  current  less  than  a 
millionth  of  an  ampere  can  be 
detected  and  measured. 

435.  Ammeters. — A  galvano- 
meter whose  scale  is  graduated  to 
read  in  amperes  is  called  an  am- 
meter (contracted  from  ampere- 
meter). Ammeters  for  industrial 
use  are  usually  of  the  D'Arson- 
val  type,  and  are  so  constructed 

that  they  can  be  carried  about 
FIG.  410.  —  D  'Arson val   Galvanometer. 

without  danger  of  injury.     The 

Weston  ammeter  (Fig.  411)  is  a  common  instrument  of  this  character. 
The  magnet  is  horizontal,  its  poles  being  at  the  narrow  side  of  the 
case,  opposite  the  scale.  The  coil  is  pivoted  on  fixed  bearings  and 


FIG.  411.  —  Weston  Ammeter. 


FIG.   412.  —  Sectional    End 
View  of  Weston  .Ammeter. 


carries  a  pointer,  B  (Fig.  412).    The  turning  effect  of  the  current  is 
opposed  by  coiled  springs,  D  and  D.     In  Fig.  412  the  nearer  pole  of 


OHM'S  LAW  537 

the  magnet  is  represented  as  partly  cut  away,  to  afford  a  better  view 
of  the  coil,  C,  and  other  interior  parts. 

An  ammeter  is  of  necessity  a  low-resistance  instrument;  for,  if 
its  resistance  were  an  appreciable  fraction  of  the  whole  resistance  of 
any  circuit  in  which  it  was  placed,  it  would  reduce  the  current  which 
it  was  placed  there  to  measure,  and  would  thus  fail  to  serve  the  in- 
tended purpose. 

V.  OHM'S  LAW 

436.  Electromotive  Force  and  potential  difference  are,  as 
a  rule,  equivalent  expressions.  They  always  mean  the  same 
kind  of  quantity;  but  usage  restricts  the  one  to  electric  cur- 
rents, while  the  other  may  refer  either  to  currents  or  to 
electric  charges. 

The  E.M.F.  of  a  cell  or  a  battery  is  the  maximum  potential 
difference  which  it  can  produce,  i.e.  the  P.D.  between  the  poles 
of  the  cell  or  the  battery  when  the  circuit  is  open  (Art.  408, 
end).  When  the  circuit  is  closed  through  a  good  conductor, 
the  poles  are  discharged  so  rapidly  that  the  P.D.  between 
them  falls  more  or  less  below  the  E.M.F.  of  the  battery,  and 
may  even  become  practically  zero;  but  the  E.M.F.  of  the 
battery  is  not  changed  unless  there  is  polarization. 

The  volt  is  the  practical  unit  of  E.M.F.  or  P.D.  As  in 
the  case  of  the  ampere,  the  pupil's  idea  of  the  volt  must 
be  gained  through  personal  acquaintance  with  the  phenom- 
ena of  electric  currents.  As  a  starting  point,  it  will  be  of 
service  to  remember  the  approximate  values  of  the  E.M.F. 
of  the  different  cells  used  in  the  laboratory  and  the  class 
room.  The  following  table  is  given  for  reference: 

ELECTROMOTIVE  FORCE  OF  CELLS  (APPROXIMATIONS) 

VOLTS  VOLTS 

Storage  cell    2.2  Dry  cell 1.4 

Bichromate  cell     2.0  Daniell   1.08 

Bunsen    1.9  Gravity 98 

Grove      1.9  Simple  voltaic  cell    98 

Leclanche"   1.4  Edison-Lalande     7 


538  ELECTRODYNAMICS 

437.  Electrical  Resistance.  —  Let  a  cell  be  connected 
with  an  ammeter  or  other  low-resistance  galvanometer 
and  the  current  measured.     Again  measure  the  current 
from  the  same  cell,  .When  it  is  sent  through  various  con- 
ductors in  turn,  e.g.  a  small  lamp,  a  piece  of  German  sil- 
ver wire,  the  coils  of  an  electromagnet,  etc.     It  will  be 
found  that,  with  any  of  these  additions  to  the  circuit,  the 
current  is  reduced  more  or  less,  probably  in  most  cases 
to  a  small  fraction  of  its  original  value. 

If  the  cell  is  one  that  does  not  polarize,  the  E.M.F. 
acting  in  all  these  circuits  is  the  same,  and  the  currents 
are  unequal  only  because  the  different  conductors  offer 
unequal  opposition  to  the  flow  of  the  current  through  them. 
This  opposition  is  a  measureable  quajitity,  and  is  called 
electrical  resistance,  or,  simply,  resistance.  With  a  given 
E.M.F.,  the  resistance  of  the  entire  circuit  (including  the 
resistance  ojLthejDell)  is^by  definition,  inversely  propor- 
tional to  the  current.  For  example,  if  the  current  is  re- 
duced one  half^when^gLConductor  is  added  to  the  circuit, 
we  know  that  this^addition  has  doubled  the  resistance  of 
the  circuit. 

The  unit  of  resistance  is  denned  as  that  resistance  through 
which  an  E.M.F.  of  one  volt  will  maintain  a  current  of  one 
ampere.  This  unit  is  called  the  ohm,  after  the  German 
physicist  Georg  Ohm.  The  ohm  is  approximately  the 
resistance  of  157  ft.  of  No.  18  copper  wire  (diameter  = 
1.024  mm.)  or  249  ft.  of  No.  16  (diameter  =  1.29  mm.). 

438.  Ohm's  Law.  —  The  current  maintained  through  a 
given  resistance  is  directly  proportional  to  the  E.M.F.  or 
the  P.D.  between  the  terminals  of  the  conductor.     This  is 
known  as  Ohm's  law,  it  having  been  discovered  experi- 
mentally by  Ohm  in  1826. 


OHM'S  LAW  539 

Ohm's  law,  when  taken  together  with  the  definition  of 
resistance,  is  stated  as  follows  :  The  current  strength  in  any 
circuit  is  directly  proportional  to  the  E.M.F.  and  inversely 
proportional  to  the  total  resistance.  This  is  one  of  the  most 
general  and  most  important  laws  of  electrical  science. 
It  holds  under  all  circumstances  for  steady  currents. 
(When  the  E.M.F.  and  the  current  are  rapidly  changing, 
other  factors  are  involved.)  < 

If  C  denotes  the  current  measured  in  amperes,  E  the 
E.M.F.  measured  in  volts,  and  R  the  resistance  of  the 
entire  circuit  in  ohms,  then  — 

C  =  |  -     (Ohm's  law)  '    .  (i) 

Ohm's  law  holds  for  any  part  of  a  circuit,  as  well  as  for  the 
entire  circuit.  Thus  if  Pbc  denotes  the  potential  differ- 
ence between  the  points  b  and  c  of  a  circuit  (Fig.  413)  and 
Rbc  the  resistance  of  the  conductor  between  these  points, 
then 


it  being  understood  that  there  is  no  cell  or  other  source 
of  E.M.F.  between  the  given  points. 

EXAMPLES.  —  The  E.M.F.  of  a  battery  of  six  storage  cells  (Fig. 
413)  is  12  volts,  and  its  resistance  is  i  ohm.  Two  lamps,  LI  and  L%, 
are  placed  in  the  circuit,  the  resistance  of  the  first  being  14  ohms  and 
that  of  the  second  9  ohms.  Find  (i)  the  current  strength,  (2)  the 
potential  difference,  P,  between  the  terminals  of  each  of  the  lamps, 
and  (3)  the  loss  of  potential  in  the  battery. 

(1)  The  total  resistance  of  the  circuit  is  i  +  14  +  9  =  24  ohms, 
assuming  that  the  resistance  of  the  connecting  wires  is  negligible; 

„      E       12 
hence  C  =  —  =  —  =  .5  ampere. 

(2)  Pbc  =  C  X  R^  =  -5  X  14  =  7  volts. 
Pde  =  C  X  #de  =  -5  X  9    =  4-5  volts. 


540  ELECTRODYNAMICS 

(3)  The  loss  of  potential  in  the  battery  is  the  same  as  it  would  be 
in  any  conductor  of  one  ohm  resistance,  when  carrying  a  current  of  .5 
ampere,  which  is  .5  X  i  =  -5  volt.  The  P.D.  between  the  poles  of  the 
battery, a  and/,  is,  therefore,  12.  -  .5  =  11.5  volts;  which,  of  course,' is 
equal  to  the  fall  of  potential  in  the  two  lamps  (7  +4.5  =  11.5). 

439.  Fall  of  Potential  along  a  Circuit.  —  In  the  above 
example  C  =  -^  =  ~^;  whence  Pbc  :  Pde  ::  Rbc  :  Rde;  or, 

-ft-bc          -tvde 

taking  the  numerical  values,  7  14.5  ::  14  19.   That  is,  the  fall 

of  potential  in  the  one  lamp 
is  to  that  in  the  other  as  the 
resistance  of  the  first  is  to 
the  resistance  of  the  second. 
This  relation  is  general.  The 
fall  of  potential  in  the  differ- 
ent parts  of  a  circuit  (except- 
ing only  the  part  or  parts 

FIG.  413-  *,.      *.  ,    * 

within  which  the  source  of 

the  E.M.F.  is  located)  is  proportional  to  the  resistance  of  the 
several  parts. 

In  that  part  of  a  circuit  in  which  the  source  of  E.M.F. 
is  located,  as  the  part  fa  in  Fig.  413,  there  is  a  rise  of  poten- 
tial, owing  to  the  chemical  action  or  other  source  of  energy. 
In  a  cell  this  rise  of  potential  takes  place  abruptly  at  the 
surface  of  the  negative  plate,  as  the  current  passes  from  the 
plate  to  the  liquid  (Art.  407).  Within  the  liquid  there  is 
a  fall  of  potential  from  the  negative  to  the  positive  plate, 
when  a  current  is  flowing,  as  in  the  case  of  any  other 
conductor. 

PROBLEMS 

1.  How  would  the  current  from  a  given  battery  be  affected  by  a  fourfold 
increase  in  the  resistance  of  the  circuit?  by  a  tenfold  increase? 

2.  What  is  the  resistance  of  a  no- volt  lamp  if  it  is  lighted  by  a  current 
of  half  an  ampere? 


LAWS   OF  RESISTANCE  541 

3.  The  E.M.F.  of  a  bichromate  cell  is  2  volts  and  its  resistance  .25  ohm. 
What  current  will  it  supply  (a)  through  an  external  resistance  of  .1  ohm? 
(6)  through  an  external  resistance  of  12  ohms? 

4.  The  E.M.F.  of  a  dry  cell  is  1.4  volts  and  its  resistance  .1  ohm.     (a) 
What  current  will  it  supply  through  an  external  resistance  of  00.5  ohm? 
(6)  What  will  then  be  the  P.D.  between  its  poles? 

6.  The  E.M.F.  of  a  gravity  cell  is  .98  volt  and  its  resistance  3.5  ohms, 
(a)  What  current  will  it  send  through  a  conductor  whose  resistance  is  negli- 
gible? (6)  What  current  will  it  send  through  an  external  resistance  of  3.5 
ohms?  (c)  What  will  be  the  P.D.  between  its  poles  in  the  first  case?  in  the 
second  case? 

VI.  LAWS  OF  RESISTANCE 

440.  Variation  of  Resistance  with  Length  and   Cross- 
section.     Experiment  shows   that  the  resistance  of  a  uni- 
form conductor  of  any  given  material  varies  directly  as  its 
length  and  inversely  as  its  cross-section. 

Equal  parts  of  a  uniform  conductor  have  equal  resistance, 
and  the  resistance  of  the  whole  conductor  is  the  sum  of  the 
resistances  of  its  parts ;  hence  the  law  of  lengths.  Similarly 
the  resistance  of  a  circuit  which  is  made  up  of  any  number 
of  different  conductors  joined  in  series  (i.e.  so  that  the  entire 
current  passes  through  each  conductor,  as  in  Figs.  405  and 
413)  is  equal  to  the  sum  of  the  resistances  of  its  parts. 

Since  the  cross-section  of  a  circular  wire  is  proportional 
to  the  square  of  its  diameter,  its  resistance  varies  inversely  as 
the  square  of  its  diameter.  Thus  if  we  take  two  wires  of 
the  same  length  and  material,  the  one  having  a  diameter 
of  i  mm.  and  the  other  a  diameter  of  2  mm.,  the  resistance 
of  the  larger  is  one  fourth  as  great  as  that  of  the  smaller, 
since  its  cross-section  is  four  times  as  great.  The  resist- 
ance of  No.  i  copper  wire,  which  is  about  the  size  of  a  lead 
pencil,  is  approximately  .62  ohm  per  mile  of  length. 

441.  Specific  Resistance.  —  Let  the  current  in  a  bat- 
tery circuit  be  measured  when  wires  of  equal  length  and 


542  ELECTRODYNAMICS 

cross-section,  but  of  different  materials,  are  included  in 
it  in  turn.  It  will  be  found  that  the  current  is  strongest 
through  copper,  considerably  weaker  through  iron,  and 
still  weaker  through  German  silver.  Evidently  the  re- 
sistance of  a  conductor  depends  upon  the  material,  as  well 
as  upon  the  length  and  cross-section. 

The  less  the  resistance  of  a  wire  of  given  length  and  cross- 
section,  the  greater  is  said  to  be  the  conductivity  of  the 
material  of  which  it  is  made  and  the  less  the  specific  re- 
sistance of  the  material.  Conductivity  and  specific  resist- 
ance are  reciprocal  quantities.  Silver  has  the  greatest 
conductivity  and  the  least  specific  resistance  of  any  known 
substance.  Copper  is  only  slightly  inferior  to  silver.  The 
following  table  gives  the  specific  resistances  of  several  com- 
mon materials,  relative  to  copper  as  the  standard.  The  value 
for  copper  is  arbitrarily  taken  as  unity.  The  values  are  to 
be  regarded  only  as  approximations,  for  the  specific  resist- 
ance of  different  specimens  of  the  same  material  is  found 
to  vary  considerably  with  the  purity  of  the  specimen,  the 
process  of  manufacture,  the  tempering,  etc. 

SPECIFIC  RESISTANCES  (RELATIVE  TO  COPPER) 

Silver,  annealed     ....  0.94  German  silver,  (varying 

Copper,  annealed  ....  i.oo  with  the  composition)    .  .  13  to  20. 

Aluminum      1.7         Manganin      33. 

Iron,  pure 6.          Mercury 59. 

Platinum    7.  Carbon,  arc  and  incandes- 

Iron,  telegraph  wire     9.  cent  lamp 2500. 

Copper  wire  is  almost  exclusively  used  for  lighting  and  power  cir- 
cuits, which  must  have  a  very  low  resistance.  Aluminum  is  the  only 
alternative  for  this  purpose,  and  is  used  to  some  extent. 

442.  Variation  of  Resistance  with  Temperature.  —  The  resist- 
ance of  metals  and  of  most  other  substances  increases  with  a  rise  of 
temperature.  The  rate  of  increase  is  nearly  the  same  for  all  pure 


LAWS  OF  RESISTANCE  543 

metals,  and  is  such  that  at  100°  C.  their  specific  resistances  are  about 
40%  higher  than  at  o°.  The  resistance  of  alloys,  particularly  man- 
ganin  and  German  silver,  is  much  less  affected  by  change  of  tempera- 
ture; hence  these  alloys  are  used  for  standard  resistance  coils  (Art. 
444).  The  resistance  of  carbon,  dilute  acids,  and  other  conducting 
solutions  decreases  with  a  rise  of  temperature.  The  carbon  filament 
of  the  common  incandescent  lamp  has  only  about  half  the  resistance 
when  white  hot  that  it  has  when  cold. 

If  a  conductor  is  maintained  at  a  constant  temperature,  its  re- 
sistance is  the  same  whatever  the  strength  of  the  current. 

443.  Laws  of  Divided  Circuits.  —  Electric  circuits  often 
have  two  or  more  branches  between  two  points,  as  between 
A  and  B  (Fig.  414).  The  branches  are 
said  to  be  connected  in  parallel,  and 
either  of  two  branches  is  called  a  shunt 
to  the  other.  This  is  the  usual  arrange- 
ment of  electric  bells  and  incandescent 
lamps  in  circuits  (Figs.  415  and  416).  FlG>  4I4* 

The  sum  of  the  currents  in  all  the  branches  between 
two  points  is  equal  to  the  current  in  the  undivided  part 
of  the  circuit.  If  the  branches  all  have  equal  resistance, 
they  take  equal  portions  of  the  current ;  if  their  resistances 
are  unequal,  the  currents  in  them  are  inversely  proportional 
to  their  resistances.  This  is  proved  for  a  two-branch  circuit 
as  follows:  Let  RI  denote  the  resistance  of  one  branch  be- 
tween A  and  B  (Fig. 
414),  and  Rz  the  re- 
sistance of  the  other ; 
and  let  Ci  and  Cz' 
denote  the  currents 

FIG.  415.  —  Electric  Lamps  m  Parallel. 

in    the    respective 

branches.  By  Ohm's  law  the  P:D.  between  A  and  B  is 
equal  to  C\R\  and  also  to  C^Rz ;  hence  CiRi  =  C2Rz,  or 


544 


ELECTRODYNAMICS 


The  resistance  between  any  two  points  of  a  circuit  is 
decreased  by  adding  one  or  more  conductors  in  parallel 
between  the  points;  for  this  is  equivalent  to  an  increase  in 
the  cross-section  of  the  original  conductor  between  the 
points.  In  the  simple  case  of  n  branches,  having  equal 
resistance,  their  combined  resistance  is  one  nth  of  the  resist- 
ance of  one  of  them;  for  the  n  branches  are  together  equiva- 
lent to  a  single  conductor  of  the  same  length  as  one  of  the 
branches  and  of  n  times  the  cross-section.  A  good  example 
is  that  of  incandescent  lamps  in  parallel  (Fig.  415). 

PROBLEMS 

1.  (a)  If  the  lamps  oh  a  no- volt  circuit  (Fig.  416)  have  each  a  resist- 
ance of  220  ohms  when  lighted,  what  is  the  joint  resistance  of  6  of 
them  in  parallel  ?  (b)  What  current  flows  in  the  leads  (the  main  wires 


C 

HH> 

a 


b  c 

FlG.  416.  —  Electric  Bell  Circuits. 


MEASUREMENT  OF  RESISTANCE  545 

of  the  circuit)  when  the  6  lamps  are  turned  on?    (c)    What  is  the  cur- 
rent when  only  one  lamp  is  lighted? 

2.  (a)  If  on  the  above  circuit  two  of  the  lamps  are  connected  in  series 
between  the  leads,  what  would  be  their  combined  resistance?     (b)  What 
would  be  the  P.D.  between  the  terminals  of  each  lamp?     (c)  What  current 
would  flow  through  them? 

3.  Describe  the  bell  circuits  shown  in  Fig.  417.     In  which  circuits  does 
only  one  bell  ring  when  one  button  is  pushed?     In  which  do  two  bells  ring 
when  a  single  button  is  pushed?     In  which  does  the  same  bell  ring  when 
either  of  two  buttons  is  pushed?     Which  circuits  require  a  larger  current 
than  circuit  a?    Which  do  not? 

4.  (a)  What  is  the  ratio  of  the  cross-sections  of  aluminum  and  copper 
wires  having  equal  resistance  per  unit  length?     (b)  What  is  the  ratio  of  the 
weight  per  unit  length  of  such  wires?     (c)  Which  material  has  the  advantage 
in  the  matter  of  conducting  power  for  a  given  size?  for  a  given  weight? 

VII.  MEASUREMENT  OF  RESISTANCE  AND  ELECTROMOTIVE 

FORCE 

444.  Standards  of  Resistance.  —  The  standard  ohm  is 
so  denned  that  it  can  be  reproduced  in  any  scientific  lab- 
oratory. It  is  the  resistance  at  o°  C.  of 
a  column  of  mercury  106.3  cm.  long  and 
i  sq.  cm.  in  cross-section.  The  mass  of 
the  mercury  is  14.4521  g.  This  standard 
is  used  only  for  comparison  in  making  FIG.  417.  —  Resistance 
and  testing  more  convenient  standards 
of  wire  for  ordinary  use.  The  latter  are  known  as  resist- 
ance coils.  They  are  made  of  some  alloy  having  a  high 
specific  resistance,  usually  manganin.  The  ends  of  each 
coil  are  joined  to  brass  blocks,  A  and  B,  B  and  C  (Fig. 
417),  arranged  in  rows  on  the  insulating  top  of  a  resistance 
box,  with  the  coils  inside  (Fig.  418).  Adjacent  blocks  are 
separated  by  a  gap,  which  is  bridged  by  means  of  a  brass 
plug.  With  all  the  plugs  firmly  in  place  the  box  resistance 
is  practically  zero;  but  wherever  a  plug  is  removed  the 


546  ELECTRODYNAMICS 

current  can  pass  only   through   the   coil   at  that  place. 

This  introduces  the  resistance  of  the  coil  into  the  circuit  of 

which  the  box  forms  a 
part.  The  amount  of 
this  resistance  is  mark- 
ed on  the  top  of  the 
box.  The  total  box 
resistance  included  in 
the  circuit  is  the  sum 
„  of  the  resistances  of  the 

FIG.  418.  —  Resistance  Box. 


are  out.  The  coils  of  a  box  make  up  the  series  .1,  .2,  .3, 
.4,  i,  2,  3,  4,  10,  20,  30,  and  40  ohms,  which  is  sometimes 
extended  to  higher  resistances. 

445.  Measurement  of  Resistance.  —  There  are  vari- 
ous methods  of  measuring  resistance.  The  practical  elec- 
trician uses  some  form  of  a  special  apparatus  known  as  a 
Wheatstone  bridge  (Lab.  Ex.  69).  The  method  of  substi- 
tution requires  only  a  resistance  box  and  any  low-resist- 
ance galvanometer.  By  this  method  the  resistance  to  be 
measured  R  (Fig.  419)  is  connected  in  circuit  with  the 
galvanometer  and  a  cell  of  constant  E.M.F.,  and  the 
deflection  is  read  as  accurately  as  possible.  The  unknown 
resistance  is  then  removed  from  the  circuit  and  the  resist- 
ance box  put  in  its  place.  The  box  resistance  is  adjusted 
to  give  exactly  the  same  deflection  as  ^-]L_^f~\G 
before.  In  making  this  adjustment,  the  f  'I 

coils  are  tried  in  order  from  larger  to    x^,^™ / 

smaller,  as  weights  are  tried  in  weighing.  R 

If  the  deflection  is  too  great,  the  resist-  Fra  4IQ- 

ance  is  too  small,  and  vice  versa.  (Why?)  The  unknown 
resistance  is  equal  to  the  box  resistance  when  the  deflec- 


MEASUREMENT  OF  RESISTANCE  547 

tions  are  equal.  This  follows  from  Ohm's  law.  For  the 
equal  deflections  indicate  equal  currents;  and,  with  a  con- 
stant E.M.F.,  the  currents  will  be  equal  only  when  the 
entire  resistance  of  the  circuit  is  the  same  in  both  cases. 
Hence  the  unknown  resistance  must  be  equal  to  the  box 
resistance  which  took  its  place. 

446.  Measurement  of  E.M.F.  and  P.D.  The  Volt- 
meter. —  In  dealing  with  electric  charges,  their  potentials 
and  potential  differences  are  measured  by  utilizing  the  elec- 
trostatic attractions  and  repulsions  of  the  charges.  The 
greater  or  less  divergence  of  the  leaves  of  an  electroscope 
can  be  made  to  serve  this  purpose.  In  dealing  with 
electric  currents,  high-resistance  galvanometers  are  used. 
The  coil  of  such  an  instrument  is  made  of  a  very  long, 
fine  wire,  and  has  a  resistance  of  hundreds  or  even  thou- 
sands of  ohms.  Owing  to  the^,  great  number  of  turns  in 
the  coil,  a  very  weak  current  catwes  a  relatively  large 
deflection.  If  the  scale  is  graduated  in  volts,  the  instru- 
ment is  called  a  voltmeter.  Voltmeters  are  usually  of  the 
D'Arsonval  type  (fixed  magnet  and  movable  coil).  An 
E.M.F. ,  when  measured 
in  volts,  is  often  called 
voltage. 

To  determine  the  P.D. 
between  two  points  of  a 
circuit,  e.g.  the  terminals 
of  the  lamp  L  (Fig.  420), 
the  voltmeter,  V,  is  con- 
nected as  a  shunt  between  FlG"  42o.-Diagram  of  Connections  for  Volt- 

meter  and  Ammeter. 

these  points.     By  Ohm's 

law,  the  current  through  the  voltmeter  is  proportional  to  the 

P.D.  between  the  points  a  and  b.    Hence,  with  a  tangent 


^^^ 
jr.] 

^^ 

b 

f 

a 

*/. 

il  fi  ii    - 

548  ELECTRODYNAMICS 

instrument,  the  P.D.  is  proportional  to  the  tangent 
of  the  angle  of  deflection.  With  instruments  of  the 
D' Arson val  type,  the  scale  can  be  graduated,  once  for 
all,  in  volts. 

To  find  the  E.F.M.  of  a  cell  or  a  battery,  the  circuit  is 
closed  through  the  voltmeter  only  (Figures  421,  422,  and 
423). 

The  necessity  for  a  high  resistance  in  the  voltmeter  arises  from 
the  fact  that,  being  connected  as  a  shunt,  it  tends  to  diminish 
the  resistance  of  the  circuit  between  the  points  with  which  it  is 
connected  (Art.  443).  But  any  appreciable  decrease  in  this  re- 
sistance would  reduce  the  P.D.  be  .ween  the  points,  since  the  fall  of 
potential  along  a  circuit  is  everywhere  proportional  to  the  resistance 
to  be  overcome  (Art.  439).  Hence  if  the  voltmeter  had  only  a  mod- 
erate resistance,  it  would  lower  the  P.D.  which  it  was  intended  to 
measure.  On  the  other  hand,  if  it  has  a  very  high  resistance,  it  does 
not  appreciably  affect  the  resistance  or  the  P.D.  between  the  points 
with  which  it  is  connected. 

447.  Measurement  of  Resistance  with  an  Ammeter  and 
a  Voltmeter.  —  If  R  denotes  the  resistance  of  a  conductor 
(in  Fig.  420,  the  lamp,  L),  and  P  the  P.D.  between  its 

ends  when  a  current,  C,  is   flowing  through  it,  then,  by 

p  p 

Ohm's  law,  C  =  ^  or  R  =  -~  (Equation   2).    Hence,  if 

the  P.D.  is  measured  with  a  voltmeter,  V,  and  the  cur- 
rent with  an  ammeter,  At  the  resistance  of  the  conductor 

p 
is  given  by  the  quotient   pr     This  is  the  simplest  and 

quickest  method  of  measuring  resistance. 

To  find  the  resistance  of  a  cell  by  this  method,  its  E.M.F. 
is  measured  with  the  voltmeter,  and  the  current  is  meas- 
ured with  the  cell  short-circuited  through  the  ammeter 
(the  resistance  of  the  ammeter  being  negligible). 


MEASUREMENT  OF  RESISTANCE  549 

448.  Arrangement  of  Cells  in  a  Battery.  —  A  battery  of 
two  or  more  cells  will,  in  general,  supply  a  larger  current 
in  a  given  circuit  than  a  single  cell  of  the  same  kind;  but 
the  current  obtained  from  a  given  number  of  cells  in  a 
given  circuit  is  largely  determined  by  the  manner  in  which 
the  cells  are  joined  together.  They  may  be  joined  in  series, 
or  in  parallel,  or  in  groups  with  both  series  and  parallel 
connections.  By  applying  Ohm's  law  we  can  readily  deter- 
mine which  arrangement  is  best  in  any  given  case.  For 
this  purpose  it  is  necessary  to  distinguish  between  the 
resistance  of  the  battery  and  the  resistance  of  the  external 
part  of  the  circuit.  The  former  is  often  called  the  internal 
resistance  and  the  latter  the  external.  Throughout  the 
present  discussion,  E  denotes  the  E.M.F.  and  R{  the  resist- 
ance of  a  single  cell,  and  Re  the  external  resistance.  Ohm's 
law,  when  applied  to  a  circuit  in  which  there  is  only  one 
cell,  then  takes  the  form  - 

C  =  "r>        ET*  (4) 

-Ki   +  Ke 

Cells  are  said  to  be  connected  in  series  when  the  entire 
current  passes  through  each  in  succession,  as  with  other 
conductors  in  series  (Fig.  421).  The  cells  are  so  joined  that 
the  E.M.F.'s  of  all  act  in  the  same 
direction  round  the  circuit,  i.e.  the  ^p  If- 
positive  pole  of  the  first  cell  is  con- 
nected with  the  negative  pole  of  the 

,    ,  .     FIG.  421.  — Cells  in  Series. 

second,  the  positive  pole  of  the  second 

with  the  negative  pole  of  the  third,  etc.  The  E.M.F.  of 
the  battery  is  the  sum  of  the  E.M.F.'s  of  the  cells,  and 
its  resistance  is  the  sum  of  the  resistances  of  the  cells,  as 
in  the  case  of  other  resistances  in  series.  With  a  battery 
of  n  like  cells  in  series  the  E.M.F.  is  nE  and  the  resistance 
nRr  Ohm's  law,  when  applied  to  a  circuit  in  which  the 


550  ELECTRODYNAMICS 

current  is  maintained  by  such  a  battery,  takes  for  them  — 

nE 

=  ^TT^e  (5) 

Cells  are  said  to  be  connected  in  parallel  when  the  circuit 
is  divided  at  the  battery,  with  a  cell  in  each  branch  (Fig. 
422).  The  negative  plates  of  all  the  cells  are  joined  to  the 
negative  terminal  of  the  battery,  N,  and  the  positive 
plates  to  the  positive  terminal,  P.  The  branches  may  all 
start  from  one  point  and  meet  at  one  point;  but  it  serves 


p 


N 

FIG.  422.  — Cells  in  Parallel. 

the  same  purpose  and  is  more  convenient  in  making  con- 
nections to  place  them  one  after  the  other,  as  shown  in  the 
figure.  This  is  exactly  like  the  connection  of  lamps  in 
parallel  between  two  leads  (Fig.  415).  All  the  negative 
plates  are  at  the  same  potential,  since  there  is  no  source 
of  E.M.F.  between  them,  and  all  the  positive  plates  are 
at  the  same  potential,  for  the  same  reason.  The  E.M.F. 
between  the  negative  and  the  positive  plates  of  the  battery 
is  that  of  one  cell  only,  or  E.  If  there  are  n  like  cells  in 
parallel,  the  resistance  of  each  being  Rb  the  resistance  of 

r>- 

the  battery  is  — »  according  to  the  law  for  conductors 
n 

in  parallel.     Hence  for  this   case  Ohm's  law  takes  the 

E 

form—  C  =  —      -•  (6) 

£  +  «, 

n 

449.   When  to  Connect  Cells  in  Series  and  when  in  Par- 
allel. —  When  cells  are  joined  in  series  the  E.M.F.  of  the 


MEASUREMENT  OF  RESISTANCE  551 

battery  varies  as  the  number  of  cells;  but  the  resistance  of 
the  battery  increases  in  the  same  ratio.  Hence,  if  the  bat- 
tery resistance  is  practically  the  whole  resistance  of  the 
circuit  (as  when  the  external  circuit  is  a  short  copper  wire), 
the  current  from  any  number  of  cells  in  series  is  not  appre- 
ciably larger  than  a  single  cell  would  supply;  i.  e.,  con- 

_f '77  TT> 

sidering  Re  negligible,  C  =  -  ^  =  •=-.     But  if  the  external 

nK\       KI 

resistance  is  relatively  large,  an  increase  in  the  battery 
resistance  has  little  effect  on  the  result  and  the  current 
increases  nearly  in  the  same  ratio  as  the  E.M.F.,  when 
cells  are  added  in  series.  A  telegraph  battery  is  a  good 
example  (Fig.  401). 

By  joining  cells  in  parallel  the  battery  resistance  is 
decreased  in  proportion  to  the  number  of  cells.  If  the 
external  resistance  is  very  small,  this  decreases  the  total 
resistance  of  the  circuit  in  nearly  the  same  ratio,  and  there 
is  a  proportionate  increase  in  the  current.  But  if  the  exter- 
nal resistance  is  large,  a  decrease  in  the  battery  resistance 
has  little  effect  on 
the  total  resistance,  j^\  '| ~\\ 
and  a  single  cell  will 
furnish  practically  as 
large  a  current  as 
any  number  of  cells 

in  parallel.  FIG.  423.  —  a,    Series-parallel,    and   b,    Parallel- 

Hence,  in  general,  series  Grouping' 

cells  should  be  joined  in  series  when  the  external  resistance 
is  relatively  high,  and  in  parallel  when  the  external  resist- 
ance is  low. 

450.    Mixed  Series  and  Parallel  Grouping.  —With  a  medium  ex- 
ternal resistance,  the  largest  current  from  a  given  number  of  cells  is 


552  ELECTRODYNAMICS 

sometimes  obtained  by  series-parallel  grouping.  The  result  is  the 
same  whether  the  cells  of  each  group  are  joined  in  series,  and  the 
groups  in  parallel  (Fig.  423,  a),  or  the  cells  of  each  group  in  parallel 
and  the  groups  in  series  (Fig.  423,  b).  The  formula  for  either  ar- 
rangement shown  in  the  figure  is  — 


^  +  RC 

2 

With  a  given  number  of  cells  and  a  given  external  resistance,  the 
largest  current  is  obtained  when  the  cells  are  so  connected  that  the 
resistance  of  the  battery  is  as  nearly  as  possible  equal  to  the  external 
resistance. 

PROBLEMS  " 

1.  What  is  the  combined  resistance  of  three  incandescent  lamps  in  par- 
allel, the  resistance  of  each  lamp  being  200  ohms? 

2.  The  fall  of  potential  through  a  coil  of  wire  is  1.5  volts  when  a  current 
of  -.2  ampere  is  flowing.     What  is  the  resistance  of  the  coil? 

3.  What  E.M.F.  will  maintain  a  current  of  1.5  amperes  through  a  re- 
sistance of  80  ohms? 

4.  If  the  E.M.F.  of  a  chromic  acid  cell  is  2  volts,  and  its  resistance  .3 
ohm,  what  current  will  it  supply  through  an  external  resistance  of  .1  ohm? 

5.  What  would  be  the  current  through  the  same  external  resistance  from 
a  battery  of  4  such  cells  (a)  in  parallel?  (6)  in  series? 

6.  What  current  would  be  supplied  by  a  battery  of  12  Leclanch6  cells, 
each  having  an  E.M.F.  of  1.4  volts  and  a  resistance  of  i  ohm,  through  an 
external  resistance  of  1.5  ohms  (a)  with  the  cells .  connected  in  series?  (6) 
in  parallel?  (c)  in  three  groups  of  four  each,  the  cells  of  each  group  being 
in  series,  and  the  groups  connected  in  parallel?     Draw  a  diagram  for  (c). 

7.  Show  that,  when  the  external  resistance  of  a  circuit  is  negligible  in 
comparison  with  the  resistance  of  a  cell,  the  current  is  porportional  to  the 
number  of  cells  connected  in  parallel,  but  a  single  cell  furnishes  as  large  a 
current  as  any  number  of  cells  connected  in  series. 

8.  Show  that,  when  the  battery  resistance  is  negligible  in  comparison 
with  the  external  resistance,  the  current  is  porportional  to  the  number  of 
cells  connected  in  series,  but  a  single  cell  furnishes  as  large  a  current  as  any 
number  of  cells  in  parallel. 


ELECTRICAL  ENERGY  553 

VIII.  ELECTRICAL    ENERGY.    HEATING    EFFECTS    OF 
ELECTRIC    CURRENTS 

451.  Electrical  Energy.  —  A  battery  or  a  dynamo  may 
be  compared  to  a  pump  which  raises  water  from  a  lower 
to  a  higher  level.  The  work  done  by  the  pump,  or  the 
energy  imparted  to  the  water,  would  be  measured  by  the 
product  of  the  weight  of  water  raised  and  the  height  to 
which  it  is  raised.  Similarly  the  function  of  a  battery  or 
a  dynamo  is  to  raise  electricity  from  a  lower  to  a  higher 
potential,  in  doing  which  it  imparts  energy  to  the  current; 
and  the  energy  is  measured  by  the  product  of  the  whole 
quantity  of  electricity  supplied  during  the  time  that  the 
current  is  flowing  and  the  potential  or  E.M.F.  at  which  it 
is  supplied.  (Compare  with  the  energy  of  an  electric 
charge,  Art.  396). 

With  a  constant  current,  the  quantity  of  electricity  sup- 
plied by  a  generator  (battery  or  dynamo)  in  a  given  time 
is  equal  to  the  product  of  the  current  strength  and  the  time ; 
just  as  the  quantity  of  water  delivered  by  a  pump  would  be 
measured  by  the  product  of  the  number  of  pounds  or  gal- 
lons per  second  and  the  number  of  seconds.  The  unit 
quantity  of  electricity  is  the  quantity  which  passes  any 
point  of  a  circuit  in  one  second  when  the  current  strength 
is  one  ampere.  This  quantity  is  called  an  ampere-second. 
(A  larger  unit  of  quantity  is  the  ampere-hour.)  A  current 
of  C  amperes,  flowing  for  t  seconds,  transports  Ct  ampere- 
seconds  of  electricity  past  every  point  of  the  circuit. 

As  stated  above,  the  electrical  energy  generated  in  a 
given  time  is  measured  by  the  product  of  the  E.M.F.  of 
the  generator  and  the  whole  quantity  of  electricity  sup- 
plied; i.e.  - 

Electrical  energy  =  volts  X  ampere-seconds. 


554  ELECTRODYNAMICS 

The  unit  of  electrical  energy  is  the  energy  imparted  to  a 
unit  quantity  of  electricity  when  its  potential  is  raised  one 
volt;  or,  a  current  of  one  ampere  generated  at  an  E.M.F. 
of  one  volt  conveys  a  unit  quantity  of  electrical  energy  in 
one  second.  This  unit  is  called  the  joule,  after  the  English 
physicist  of  that  name  (Arts.  242  and  454).  Hence,  if 
E  denotes  the  E.M.F.  of  the  generator  in  volts,  C  the  num- 
ber of  amperes,  and  /  the  number  of  seconds,  the  quantity 
of  electrical  energy  generated  in  that  time  is  ECt  joules; 
or,  briefly  - 

Electrical  energy  =  ECt  joules.  (7) 

The  equivalent  of  one  joule  in  mechanical  energy  is 
.74  ft.-lb.,  approximately. 

452.  Energy  Expended  by  an  Electric  Current.  —  The 

energy  imparted  to  an  electric  current  by  the  generator 
is  expended  (changed  into  other  forms  of  energy)  in  the 
circuit.  It  is  transformed  (i)  into  heat  in  overcoming 
the  resistance  of  the  circuit,  as  in  the  electric  lamp,  (2)  into 
chemical  energy  in  producing  chemical  change,  as  in 
charging  a  storage  battery,  and  (3)  into  mechanical  energy 
in  doing  mechanical  work,  as  in  running  electric  motors. 

The  work,  of  whatever  kind,  done  by  a  current  in  the 
different  parts  of  a  circuit  is  everywhere  proportional  to 
the  fall  of  potential.  Thus  if  the  P.D.  between  any  two 
points,  a  and  b,  of  the  circuit  is  denoted  by  Pab,  the  work 
done  by  the  current  between  these  points  in  /  seconds  is 
PabO  joules,  or,  briefly,  — 

Work  done  between  a  and  b  =  PabCt  joules.        (8) 

453.  Electrical  Power.     Industrial  Units  of   Electrical 
Power  and  Energy.  —  The  power  of  an  agent,  as  defined  in 
the  study  of  Mechanics  (Art.  136),  is  its  rate  of  doing  work. 


ELECTRICAL  ENERGY  555 

The  power  of  a  battery  or  a  dynamo  is  the  rate  at  which  it 
generates  electrical  energy,  and  is  measured  by  the  energy 
generated  in  one  second,  that  is  — 

Electrical  power  =  EC  joules  per  second.          (9) 

One  joule  of  work  per  second  is  a  unit  of  power,  and  is 
called  a  watt,  after  James  Watt,  the  inventor  of  the  modern 
steam  engine.  Since  this  unit  is  very  small,  a  unit  1000 
times  as  large,  called  the  kilowatt,  is  adopted  for  industrial 
use.  Dynamos  and  motors  are  very  generally  rated  in 
kilowatts.  A  watt  is  equal  to  yjg  °f  a  horse-power,  or 
to  .74  foot-pound  per  second.  A  kilowatt  is  equal  to 
"We0"  horse-power,  or  f  horse-power,  very  nearly.  In 
terms  of  these  units, — 

Electrical  power  =  EC  watts,  (10) 

EC  kilowatts,  (n) 

C  horse-power.  (12) 

The  power  expended  in  any  part  of  an  electric  circuit 
is  proportional  jointly  to  the  fall  of  potential  in  that  part 
and  to  the  strength  of  the  current,  or 

Power  expended  between  a  and  b  =  PabC  watts.      (13) 

For  example,  if  a  no- volt  lamp  takes  a  current  of  .5  ampere, 
the  power  consumed  in  lighting  it  is  110  X  .5  =  55  watts,  or 

— -2  =  —  horse-power,  nearly. 
746       14 

The  industrial  units  of  electrical  energy  are  the  watt-hour  and  the 
kilowatt-hour.  The  watt-hour  is  the  energy  expended  in  one  hour  at 
the  rate  of  one  watt,  or  one  joule  per  second;  hence  it  is  equal  to  3600 
joules.  The  kilowatt-hour  is  equal  to  1000  watt-hours.  The  cost 
of  electrical  energy  to  consumers  for  light  and  other  uses  is  generally 
between  7  cents  and  10  cents  per  kilowatt-hour. 

An  instrument  for  measuring  the  power  of  a  current  is  called  a 
watt-meter.  A  watt-hour-meter  records  on  a  set  of  dials  the  number 
of  watt-hours  of  energy  consumed.  In  this  instrument  the  current 


556  ELECTRODYNAMICS 

drives  a  small  motor,  so  designed  that  its  speed  is  proportional 
jointly  to  the  number  of  volts  and  the  number  of  amperes.  The 
motor  drives  a  train  of  gear-wheels;  and  the  wheels  move  the  hands 
of  the  dials,  as  in  the  gas  meter. 

/ 
454.   Heat  Generated  in  a  Conductor.     Joule's  Law.  - 

When  a  current  of  several  amperes  is  sent  through  a  piece 
of  fine  German  silver  or  platinum  wire,  the  wire  becomes 
white  hot,  and  may  even  melt;  but  the  remainder  of  the 
circuit,  if  it  consists  of  larger  copper  wire,  is  not  appre- 
ciably warmed. 

Electrical  energy  is  converted  into  heat  in  overcoming 
the  resistance  of  the  circuit.  If  no  other  work  is  done  by 
the  current,  all  its  energy  is  thus  transformed.  Since  all 
conductors  have  resistance,  all  are  heated  more  or  less 
when  carrying  a  current ;  but  with  good  conductors,  having 
a  size  adapted  to  the  strength  of  the  current,  the  amount 
of  heating  is  slight  and  generally  passes  unnoticed. 

Since  the  fall  of  potential  in  the  different  parts  of  a  cir- 
cuit is  proportional  to  the  resistances  of  the  parts  (Art. 
439),  and  since  the  loss  of  electrical  energy  is  proportional 
to  the  fall  of  potential  (Formula  13),  it  follows  that  the  heat 
generated  in  the  different  parts  of  a  circuit  (in  all  of 
which  parts  the  same  current  is  flowing)  is  proportional 
to  their  resistances.  The  German  silver  or  platinum  wire 
in  the  experiment  described  above  becomes  very  hot,  while 
the  copper  connecting  wires  remain  cool,  because  its  resist- 
ance for  a  given  length  is  many  times  greater  than  that 
of  the  copper  wires.  Electric  lighting  furnishes  another 
example  of  the  same  conditions.  The  circuit  wires  are  of 
copper,  large  enough  to  carry  the  current  without  noticeable 
heating;  but  the  lamp  filaments  have  a  high  resistance. 
Hence  the  fall  of  potential  and  the  conversion  of  electrical 
energy  into  heat  take  place  almost  wholly  in  the  lamps. 


ELECTRICAL  ENERGY 


557 


Since  the  energy  converted  into  heat  in  a  conductor  is  equal  to 
P.D.  X  Ct  (Formula  8),  and  since  P.D.  =  CR,  R  being  the  resist- 
ance of  the  conductor  (Ohm's  law),  it  fol- 
lows that  the  heat  generated  is  equal  to 
C?Rt  joules.  In  order  to  express  this 
quantity  of  heat  in  terms  of  the  custom- 
ary heat  unit,  the  calorie,  it  is  necessary 
to  know  the  equivalent  of  the  joule  in 
calories.  This  is  found  by  experiment 
to  be  .24  calorie;  hence 

Heat  generated  =  .z^C-Ri  calories.  (14) 
This  is  known  as  Joule's  law,  it  having 


FIG.  424. 


been  experimentally  determined  by  him  in  1841.     The  experiment 
consists  in  passing  a  known  current  for  a  measured  length  of  time 
through  a  coil  of  known  resistance,  placed  in  a  calorimeter  contain- 
ing water   (Fig.  424).     The  electrical  energy 
converted  into  heat  in  the  calorimeter  is  equal 
to  C2Rt  joules.     The  heat  generated  is  com- 
puted in  calories  from  the  mass  of  the  water, 
the  mass  and  specific  heat  of  the  vessel,  and 
the  rise  of  temperature. 

455.   Incandescent    Lamps.  —  In    an 

incandescent  lamp  (Fig.  425),  the  current 
passes  through  a  slender  filament  or  wire 
which,  owing  to  its  high  resistance,  is 
heated  white  hot.  The  ends  of  the  fila- 
ment are  attached  to  short  platinum 
wires,  which  pass  through  the  glass  and 
connect  with  the  metal  casing,  d,  and 
plug,  g,  at  the  base  of  the  lamp.  When 

FIG.  425.  —  Diagram  of  the  lamp  is  screwed  into  the  socket,  g  is 
Ele°"  brouSnt  into  contact  with  h;  the  circuit 
is  closed  at  c  by  turning  the  key  x.  The 

air  is  exhausted  from  the  bulb  to  prevent  combustion  of 

the  filament  when  heated. 


558  ELECTRODYNAMICS 

In  the  earliest  type  of  incandescent  lamp,  which  is  still 
the  most  common,  the  filament  is  of  carbon.  These  lamps 
are  made  for  both  no- volt  and  2  20- volt  circuits,  and  are 
of  various  candle  powers,  the  i6-candle  lamp  being  in  most 
general  use.  The  i6-candle,  no- volt  lamp  has  a  resistance 
of  about  220  ohms  when  lighted,  and  hence  takes  a  cur- 
rent of  about  half  an  ampere.  The  power  consumed  is 
no  X  .5  =  55  watts.  The  i6-candle,  220- volt  lamp  also 
consumes  55  watts,  and  hence  takes  a  current  of  .25  am- 
peres (55  -z-  220  =  .25).  By  Ohm's  law  the  resistance  of 
the  heated  filament  is  220  -f-  .25  =  880  ohms.  To  secure 
this  high  resistance,  the  filament  is  made  longer  and  of 
smaller  size  than  that  of  the  no- volt  lamp. 

The  newer  types  of  incandescent  lamps  have  metallic 
filaments.  None  of  the  common  metals  or  alloys  would 
serve  for  this  purpose,  since  their  melting  points 
are  all  too  low.  The  best  results  have  been  ob- 
tained with  tungsten  and  tantalum.  As  the  spe- 
cific resistance  of  these  materials  is  much  less 
than  that  of  carbon,  the  filaments  must  be  long 
and  very  slender  even  for  iio-volt  circuits. 
The  advantage  of  the  metallic-filament  lamps 
is  their  high  efficiency,  or  large  candle  power 
for  the  number  of  watts  consumed.  In  this 
resPect  tne  tungsten  lamp  (Fig.  426)  ranks  first, 
requiring  only  1.25  watts  per  candle  power,  while 
the  ordinary  carbon-filament  lamp  requires  3.5  watts  per 
candle  power  (55  ^-  16  =  3.5). 

466.  The  Electric  Arc  and  the  Arc  Lamp.  —When  the 
rounded  ends  of  two  pieces  of  carbon  loosely  touch  each 
other  and  a  current  of  several  amperes  passes  between 
them,  the  ends  quickly  become  white  hot.  The  heating  is 
due  to  the  relatively  high  resistance  where  the  surfaces 


ELECTRICAL  ENERGY  559 

make  imperfect  contact  with  each  other.     If  the  source  of 

the  current  is  a  dynamo  or  a  storage  battery,  capable  of 

maintaining  a  P. D.  of  40  or  50  volts 

between  the  carbons  when  they  are 

separated  slightly,  the  current  will  be 

conducted  across  the  gap  by  the  heated 

air  and  carbon  vapor.     The  luminous 

track  of  the  current  through  the  air  is 

known  as  the  electric  arc  (Fig.  427). 

The  greater  part  of  the  light  comes 

from  the  carbon  points,  especially  from 

the  depression  or  crater,  which  forms  at 

the  end  of  the  positive  carbon. 

The  carbon  rods  of  an  arc  lamp  burn  FlG'  427'  ~  Electric  Arc' 
away  more  or  less  rapidly;  and,  if  they  were  held  in  a  fixed 
position,  the  arc  would  increase  in  length  and  finally  go  out, 
when  the  resistance  of  the  gap  became  too  great  for  the  cur- 
rent to  pass.  Arc  lamps  are  therefore  provided  with  auto- 
matic devices  of  various  sorts,  by  means  of  which  the  upper 
carbon  is  permitted  to  drop  and  touch  the  lower  one,  for  the 
purpose  of  starting  the  arc,  and  is  thereafter  lowered  a  little 
from  time  to  time  as  the  carbons  burn  away.  This  "feed- 
ing" mechanism,  as  it  is  called,  is  controlled  by  the  current 
which  lights  the  lamp,  through  the  action  of  electromag- 
nets. 

The  regulating  device  shown  in  Fig.  428  consists  of  two  magnet 
coils,  Ci  and  Ci,  with  their  movable  iron  cores,  n  and  n,  a  lever,  /, 
and  a  clutch,  c.  C\  is  a  low-resistance  coil,  in  series  with  the  carbons; 
C2  is  a  high-resistance  coil,  connected  as  a  shunt  to  the  carbons. 
When  there  is  no  current  flowing  in  the  circuit,  the  clutch  releases 
the  metal  rod  to  which  the  upper  carbon  is  fastened,  and  the  rod 
drops  till  the  carbons  meet.  When  the  current  is  turned  on,  prac- 
tically the  whole  of  it  passes  by  way  of  the  carbons  and  the  coil  C\. 
The  magnetic  pull  of  this  coil  draws  the  core  n  up,  thus  lifting  the 


560 


ELECTRODYNAMICS 


right  end  of  the  lever  /.     This  motion  is  communicated  to  the  clutch, 
which  grips  the  rod  and  raises  it  slightly;   the  rod  lifts  the  upper 

carbon,  and  the  arc  is  established. 
As  the  carbons  burn  away,  the  resist- 
ance of  the  arc  increases,  the  current 
across  the  arc  and  through  the  coil  Ci 
decreases,  and  an  increasing  fraction 
of  the  current  flows  through  the  shunt 
coil  C%.  This  goes  on  until  the  mag- 
netic pull  of  C2  is  sufficient  to  raise 
the  core  n.  As  n  rises,  it  momentarily 
releases  the  clutch,  and  the  rod  falls  a 
short  distance,  bringing  the  carbons 
closer  together. 

The  electric  arc  requires  a  P. D. 
of  45  to  50  volts.  The  current 
varies  with  the  diameter  of  the 
carbons.  Street  lamps  of  the 
usual  size  take  about  10  am- 
peres. These  give  a  light  of 
about  1000  candle  power  in  the 
direction  of  greatest  intensity.  The  average  for  all  direc- 
tions in  space  about  the  lamp  (called  the  "mean  spherical" 
candle  power),  lies  between  300  and  500  candle  power. 
The  arc  lamp,  therefore,  takes  approximately  one  watt  per 
candle  power. 

There  are  several  types  of  arc  lamps.  The  enclosed  arc  receives 
its  name  from  the  fact  that  it  is  completely  inclosed  in  a  small  globe. 
Although  this  is  not  air  tight,  it  practically  excludes  fresh  air;  and  the 
carbons  last  much  longer  than  they  do  with  an  open  arc.  In  the  flam- 
ing arc  lamp  the  carbons  have  a  central  core,  filled  with  lime  and 
other  minerals  which  change  the  arc  to  an  intensely  brilliant  flame. 
The  flaming  arc  has  the  highest  efficiency  of  any  artificial  source  of 
light,  requiring  only  .6  watt  per  mean  spherical  candle  power.  There 
are  various  other  forms  of  electric  lamps,  and  the  number  is  increas- 
ing from  year  to  year. 


FIG.  428.  —  Diagram  of  Arc  Lamp. 


ELECTRICAL   ENERGY  561 

457.  Electric  Forging  and  Smelting.  —  The  temperature  of  the 
electric  arc  is  estimated  at  3500°  to  3800°  C.,  and  is  the  highest  that 
can  be  produced  by  any  known  means.     At  this  temperature  the  most 
refractory  substances  are  melted  and  vaporized.    The  heat  of  the 
arc  is  utilized  in  the  reduction  of  ores  and  in  the  manufacture  of  nu- 
merous chemical  products,  e.g.  calcium  carbide,  carborundum,  and 
graphite.     Such  processes  are  conducted  on  a  large  scale  at  Niagara 
Falls  and  other  places,  where  water-power  is  abundant  and  can  be 
cheaply  converted  into  electrical  power.     The  materials  to  be  treated 
are  placed  in  a  huge  furnace,  and  a  current  of  hundreds  or  thousands 
of  amperes  is  passed  through  the  mass  between  large  carbon  elec- 
trodes. 

In  recent  years  the  electric  current  has  largely  superseded  the 
forge  fire  as  the  source  of  heat  for  welding,  brazing,  shaping,  and  tem- 
pering metals.  The  heat  is  generated  within  the  metal  by  sending 
through  it  a  very  large  current  at  a  low  voltage.  In  welding,  the 
two  pieces  of  metal  are  heated  in  this  manner  while  pressed  firmly 
together.  When  they  have  become  soft,  they  are  squeezed  together 
slightly,  the  current  is  shut  off,  and  the  weld  is  complete. 

458.  Cooking  and   Heating  by  Electricity.  —  Heat  is  obtained 
from  electricity  for  cooking  and  other  uses  in  the  home  by  sending  the 
current  through  coils  of  suitable  resistance.     The  electric  flat-iron  is 
perhaps  the  most  familiar  example.     The  resistance  coil  is  within  the 
hollow  body  of  the  iron,  just  above  the  bottom 

plate  (Fig.  429).     A  layer  of  asbestos,  placed 
over  the  coil,  largely  prevents  the  conduction 
of  heat  upward.     The  heating  coil  of  cooking 
utensils  is  either  within  the  vessel  or  directly 
underneath  it.     Electric  cooking  and  heating 
appliances  are  of  ten  connected  to  lamp  sockets; 
but  it  should  be  borne  in  mind  that  they  take  a 
much  greater  current  than  a  lamp  does.    Flat- 
irons,  chafing-dishes,  coffee  urns,  etc.,  take 
from  250  to  500  watts,  or  as  much  as  5  to          ance  Ribbon. 
10  carbon-filament  lamps;  and  the  larger  ap- 
pliances, such  as  stoves,  ovens,  and  broilers,  take  from  500  to  1800 
watts.     Now  the  wire  of  lamp  circuits  is  not  large  enough  to  carry 
more  than  a  few  amperes  without  overheating,  which  destroys  the 


562  ELECTRODYNAMICS 

insulation  and  may  set  the  house  on  fire.  Hence  in  making  such 
use  of  lamp  circuits  one  must  be  careful  to  avoid  an  "  overload  "  of 
current.  The  wiring  for  an  "  electric  kitchen  "  is  of  much  larger  size 
than  that  regularly  used  for  lighting  circuits. 

The  electric  current  is  a  cleanly,  safe,  and  efficient  source  of  heat 
in  the  home.  The  one  obstacle  in  the  way  of  -its  extensive  use  for 
this  purpose  is  the  expense,  which,  so  far  as  the  general  heating  of 
dwellings  is  concerned,  is  prohibitive.  The  output  of  electrical  energy 
from  a  modern  power  station  is,  on  the  average,  not  more  than 
10%  of  the  energy  of  the  fuel  consumed  in  producing  it.  And  this 
small  fraction  is  obtained  at  great  expense;  so  that,  in  the  end,  a  given 
amount  of  heat  from  the  electric  current  costs  the  consumer  probably 
fifty  times  as  much  as  an  equal  amount  obtained  directly  from  fuel. 
On  the  other  hand,  the  heat  of  the  current  can  be  put  where  it  is 
wanted  and  generated  only  as  long  as  it  is  wanted;  hence  very  little 
is  wasted,  and  for  light  cooking,  ironing,  and  similar  uses,  the  cost  is 
not  excessive. 

459.  Safety  Fuses.  —  In  the  use  of  electricity  for  light- 
ing and  power  purposes,  there  is  always  the  possibility 
that  the  mains  or  leads  may  be  accidentally  connected 
by  a  conductor  of  low  resistance,  or  short-circuited.  If 
this  happens,  the  current  instantly  increases  perhaps  a 
hundred  or  a  thousandfold;  for  the  resistance  of  the  circuit 
is  then  only  a  few  ohms  at  the  most.  With  such  a  current 
the  wires  quickly  become  hot,  and  are  likely  to  set  fire  to 
woodwork  and  other  inflammable  materials  near  them 
before  the  trouble  is  discovered  and  remedied.  As  a  pro- 
tection against  the  danger  of  a  short-circuit  and  the  lesser 
danger  of  an  overload  in  ordinary  use,  safety  fuses  are 
placed  at  suitable  points  in  all  lighting  and  power  circuits. 

Fuse  wire  is  made  of  lead  and  tin,  or  other  alloy  which  melts  at 
a  low  temperature.  It  is  sometimes  used  as  a  bare  wire;  but  in  reg- 
ular practice  it  is  in  the  form  of  an  inclosed  fuse  (Fig.  430).  This 
consists  of  a  fuse  wire  inclosed  in  a  fiber  tube,  which  is  filled  in  with 
some  non-conducting,  infusible  material.  The  wire  is  joined  to  brass 


c 


ELECTRICAL  ENERGY  563 

caps  which  cover  the  ends  of  the  tube;  and  the  caps  connect  with  the 
circuit  through  the  spring  clips  by  which  they  are  held  in  place  on 
the  fuse  block.  The  latter  is  of  por- 
celain to  prevent  the  possibility  of 
fire  when  a  fuse  "  blows  out." 

Fuses  are  of  different  sizes,  vary- 
ing with  the  number  of  amperes 
which  they  are  designed  to  carry. 
The  fuses  in  any  circuit  should  have 
a  capacity  just  above  the  maximum  a  b 
which  the  circuit  should  carry.  Then,  FIG.  430.  —  Fuses,  a,  Single  Fuse; 
if  for  any  reason  the  current  exceeds  b>  the  Same  in  Position;  c, 
*  Fuse  Block  for  three  Circuits, 

this  maximum,  the  fuse  wire  will  in- 
stantly melt,  breaking  the  circuit,  and  no  further  damage  will  be 
done. 

PROBLEMS 

1.  What  is  the  power  of  a  battery  that  is  able  to  maintain  a  current  of 
4  amperes  through  a  resistance  of  6  ohms? 

2.  (a)  Assuming  equal  efficiency,  how  does  the  current  taken  by  a  50- 
candle  lamp  compare  with  that  taken  by  a  i6-candle  lamp  for  the  same 
voltage?      (6)  How  does  the  resistance  of  the  5o-candle  lamp  compare  with 
that  of  the  other? 

3.  An  electric  oven  takes  1500  watts  and  a  chafing-dish  500  watts  on  a 
1 10- volt  circuit.     What  is  the  resistance  of  the  coil  in  each? 

4.  What  is  the  relation  between  the  resistance  of  a  coil  or  a  lamp  and  the 
heat  generated  in  it  per  second  on  a  constant- potential  circuit? 

6.   What  is  the  relation  between  the  resistance  of  a  coil  and  the  heat 
generated  in  it  per  second  on  a  constant-current  circuit? 

6.  What  power  will  be  required  to  light  225  tungsten  lamps  of  32  candle 
power,  the  efficiency  being  1.25  watts  per  candle  power? 

7.  If  in  lighting  the  above  lamps  10%  of  the  power  generated  by  the 
dynamo  is  lost  in  the  distributing  wires,  what  must  be  the  power  of  the 
dynamo  in  kilowatts? 

8.  If  the  above  dynamo  is  run  by  a  turbine  water-wheel  having  an  effi- 
ciency of  80%,  what  water-power  will  be  required  to  run  the  wheel?     Express 
the  result  in  horse-power. 

9.  At  9  cents  per  kilowatt  hour,  what  will  it  cost  to  heat  to  the  boiling 
point  4  liters  (a  little  more  than  i  gal.)  of  water,  taken  at  20°  C.,  assum- 
ing that  75%  of  the  heat  generated  goes  to  the  water? 


564  ELECTRODYNAMICS 

IX.  ELECTROMAGNETIC  INDUCTION 

460.  Historical  Note.  —  The  generation  of  electric  cur- 
rents by  magnetic  action,  as  in  the  dynamo,  is  termed 
electromagnetic    induction,    or    current    induction.     The 
phenomena  of  current  induction  were  first  observed  and 
studied  by  Joseph  Henry  in  America  and  Michael  Fara- 
day in  England.     Knowing  that  an  electric  current  affects 
a  magnet,  Faraday  reasoned  that  a  magnet  should  react 
upon  a  current.     Working  from  this  hypothesis,  he  tried 
again  and  again  during  a  period  of  several  years  to  dis- 
cover such  an  effect,  and  at  last,  in  1831,  his  efforts  were 
rewarded  with  success  —  he  had  discovered  electromagnetic 
induction.     The  importance  of  this  discovery  could  not 
have  been  foreseen  at  the  time.     Faraday  himself  was  so 
far  from  suspecting  it  that  he  says  in  a  letter,  "I  am  busy 
just  now  again  on  electromagnetism,  and  I  think  I  have 
got  hold  of  a  good  thing,  but  can't  say.     It  may  be  a  weed 
instead  of  a  fish  that,  after  all  my  labor,  I  may  at  last  pull 
up."    He  had  indeed  got  hold  of  a  good  thing.     Years 
afterward  Tyndall  wrote:  "I  can  not  help  thinking  that 
this   discovery  is   the   greatest   experimental   result   ever 
obtained.     It  is  the  Mont  Blanc  of  Faraday's  own  achieve- 
ments.    He  always  worked  at  great  elevations,  but  higher 
than  this  he  never  attained."     While  great  honor  is  due  to 
Faraday  for  this  most  useful  contribution  to  science,  Henry's 
name  will  also  be  remembered.     His  discoveries  in  part 
preceded  Faraday's,  but  he  did  not  publish  them  till  later. 

461.  Current  Induction  by  a  Magnet.  —  Induced  cur- 
rents can  be  generated  in  various  ways,  but  these  are  only 
different  ways  of  bringing  about  the  one  essential  condition, 
namely,  a  change  in  the  magnetic  field  within  a  closed  circuit. 
This  change  may  be  either  an  increase  or  a  decrease  in 


ELECTROMAGNETIC  INDUCTION  565 

the  strength  of  the  field,  or  a  change  in  the  direction  of  the 
lines  of  force  relative  to  the  circuit. 

The  induced  currents  obtained  in  experiments  are  usu- 
ally very  weak;  and,  in  order  to  increase  the  induction,  a 
long  wire  in  the  form  of  a  coil  of  several  hundred  turns  is 
taken  for  the  circuit  (Fig.  431).  This  coil  is  joined  with  a 
sensitive  galvanometer,  which  serves  to  show  the  presence 
of  a  current  and  to  determine  its  direction.  With  such 
a  circuit  and  a  strong  magnet,  the  laws  of  current  induc- 
tion can  be  readily  determined. 

Let  the  magnet  be  thrust  quickly  into  the  coil,  and,  after 
a  few  seconds,  quickly  removed.  The 
galvanometer  indicates  a  momentary 
current  in  one  direction  while  the  mag- 
net is  being  inserted,  and  in  the  opposite 
direction  while  it  is  being  withdrawn; 
but  there  is  no  current  while  the  mag- 
net remains  at  rest  within  the  coil.  The 
field  of  the  magnet  induces  a  current 
not  by  'its  presence  merely,  but  by  its 
motion  relative  to  the  coil,  as  the  magnet 
is  moved.  As  stated  above,  this  is  FIG.  431.  — Current  in- 

..  .     .      .  .  ...          duction  by  Magnet. 

only  one  of  several  devices  by  which 

the  strength  of  the  magnetic  field  within  a  circuit  may  be 

changed. 

462.  Lenz's  Law.  —  In  the  above  experiment  the  direc- 
tion of  the  current  through  the  galvanometer  is  indicated 
by  the  direction  of  the  deflection.  Starting  from  the  gal- 
vanometer, we  can  trace  the  direction  of  the  current  round 
the  coil;  and  we  can  then  find  the  polarity  of  the  coil  by 
applying  the  right-hand  rule.  Proceeding  thus,  we  find 
that  the  nearer  (upper)  end  of  the  coil  is  N  while  the  N 


566 


ELECTRODYNAMICS 


FIG.  432.  —  Direction  of  In- 
duced Current. 


pole  of  the  magnet  is  being  inserted,  and  is  S  while  the 
N  pole  is  being  withdrawn  (Fig.  432).     Inserting  the  S 

pole  of  the  magnet  makes  the 
nearer  end  of  the  coil  S;  re- 
moving it  makes  this  end  N. 

Thus  while  either  pole  of  the 
magnet  is  being  inserted,  the 
motion  is  opposed  by  the  repul- 
sion of  the  nearer  (like)  pole  of 
the  coil;  and  while  either  pole 
is  being  withdrawn  the  motion 
is  opposed  by  the  attraction  of 
the  nearer  (unlike)  pole  of  the  coil.  Hence,  in  general, 
"  The  induced  current  is  in  such  a  direction  as  to  oppose  by 
its  electromagnetic  action  the  motion  of  the  magnet  or  the  coil 
(see  Art.  463)  which  produces  the  induction."  This  is  known 
as  Lenz's  law. 

To  move  the  magnet  against  the  opposing  magnetic 
forces  requires  an  expenditure  of  mechanical  energy,  which 
becomes  the  electrical  energy  of  the  induced  current. 
The  amount  of  energy  thus  transformed  in  the  experiment 
is  exceedingly  small;  but  the  process  is  of  the  same 
nature  as  that  which  takes  place  in  all  dynamos,  and  in 
the  largest  is  capable  of  generating  electrical  energy  at 
the  rate  of  several  thousand  horse-power.  The  principle 
involved  is  therefore  of  the  greatest  importance. 

Lenz's  law  is  only  a  special  case  under  the  general  prin- 
ciple of  the  conservation  of  energy.  If  the  direction  of 
the  induced  current  were  such  that  its  magnetic  action 
aided  the  motion  which  produces  the  current,  a  dynamo 
would  run  of  itself  when  once  started,  and  generate  elec- 
trical energy  out  of  nothing;  or,  in  other  words,  it  would 
be  a  "  perpetual-motion  machine." 


ELECTROMAGNETIC  INDUCTION  567 

463.  Current  Induction  by  a  Current.  —  Results  sim- 
ilar to  the  above  are  obtained  when  a  long,  slender  coil,  in 
which  a  current  is  flowing,  takes  the  place  of  the  magnet 
in  the  experiment.  This  coil  is  called  the  primary  coil, 
and  its  current  the  primary  or  inducing  current.  The 
larger  coil  is  known  as  the  secondary  coil,  and  the  current 
induced  in  it  is  often  called  the  secondary  current.  The 
primary  current  is  supplied  by  a  battery. 

If  the  primary  coil  is  small  enough  to  go  inside  the  sec- 
ondary, the  results  obtained  when  it  is  inserted  or  with- 
drawn are  the  same  as  in  the  corresponding  case  with  the 
magnet.  Thus  if  the  lower  end  of  the  primary  coil  is  N, 
thrusting  it  in  induces  a  current  which  makes  the  upper 
end  of  the  secondary  coil  N.  Hence  in  this  case  the  direc- 
tion of  the  induced  current  is  opposite  to  that  of  the  pri- 
mary current,  and  it  is  therefore  called  an  inverse  induced 
current.  A  direct  induced  current  is  one  whose  direc- 
tion round  the  coil  is  the  same  as  that  of  the  primary 
current.  Experiment  shows  that  the  induced  current  is 
inverse  when  either  pole  of  the  primary  coil  is  inserted, 
and  direct  when  it  is  withdrawn.  It  will  be  useful  to 
remember  that,  to  an  observer  looking  in  the  direction  of 
the  lines  of  force  of  the  inducing  magnetic  field,  an  inverse 
induced  current  flows  counter-clockwise  round  the  coil,  and 
a  direct  induced  current  clockwise. 

If  the  primary  circuit  is  closed  or  broken  while  the  pri- 
mary coil  remains  at  rest  within  the  secondary,  the  induc- 
tion is  the  same  as  when  the  primary  coil  is  inserted  or 
removed  with  the  current  flowing;  for  this  is  only  another 
way  of  changing  the  magnetic  field  within  the  secondary. 

The  induced  currents  are  in  all  cases  much  stronger  when 
the  primary  coil  contains  a  soft  iron  core;  for  the  iron  greatly 
increases  the  strength  of  the  magnetic  field. 


568  ELECTRODYNAMICS 

A  review  of  all  the  cases  considered  will  show  that  the 
direction  of  the  induced  current  is  given  by  the  following 
general  law:  An  increase  in  the  strength  of  the  magnetic  field 
within  a  closed  circuit  induces  an  inverse  current,  and  a 
decrease  in  the  strength  of  the  field  induces  a  direct  current. 
These  directions  are  such  that  the  magnetic  action  of  the 
induced  current  is  always  in  agreement  with  Lenz's  law. 

464.  Magnitude  of  the  Induced  E.M.F.  —  An  induced 
current  is  due  to  an  induced  E.M.F.,  and,  in  a  given  cir- 
cuit, is  proportional  to  it.  The  E.M.F.  induced  in  a  given 
circuit  is  proportional  to  the  rate  of  increase  or  decrease  of 
the  magnetic  field  within  the  circuit.  This  can  be  shown 
qualitatively  by  varying  the  speed  with  which  the  primary 
coil  or  the  magnet  is  thrust  into  the  secondary.  With  the 
primary  coil  and  an  iron  core,  the  deflection  of  a  sensitive 
galvanometer  is  very  large  when  the  motion  is  rapid;  but 
it  becomes  less  and  less  indefinitely  as  the  coil  is  moved 
more  and  more  slowly.  With  a  given  pair  of  coils  and  a 
given  primary  current,  the  greatest  possible  E.M.F.  is  in- 
duced by  breaking  the  primary  circuit  while  the  primary 
coil  and  the  iron  core  are  at  rest  within  the  secondary;  for 
the  magnetic  field  within  the  secondary  is  very  strong  to 
begin  with,  and  breaking  the  circuit  removes  it  in  the  quick- 
est possible  way.  The  induced  E.M.F.  at  "break"  is 
so  great,  even  with  coils  of  ordinary  size,  that  a  distinct 
shock  is  received  from  the  secondary  when  its  terminals 
are  touched  with  the  fingers. 

Other  conditions  remaining  the  same,  the  induced  E.M.F. 
is  proportional  to  the  number  of  turns  in  the  secondary  coil. 
For  any  change  in  the  magnetic  field  induces  an  E.M.F. 
in  each  turn,  just  as  if  the  other  turns  were  not  present. 
Primary  coils  are  commonly  made  with  a  relatively  small 


ELECTROMAGNETIC  INDUCTION  569 

number  of  turns  (from  100  to  200),  and  secondary  coils 
with  many  thousand  turns.  With  such  coils,  the  induced 
E.M.F.  is  relatively  high,  as  indicated  by  the  fact  that 
shocks  may  be  obtained  from  them.  If  the  primary  cur- 
rent is  sent  through  the  larger  coil  and  the  smaller  one  is 
used  as  the  secondary,  the  induced  E.M.F.  will  be  low. 

465.  Self-induction.  —  Joseph  Henry  seems  to  have  been  the 
first  to  observe  that  a  brilliant  spark  occurs  when  a  circuit  containing 
the  coil  of  a  large  electromagnet  is  broken.  If  one  end  of  such  a  cir- 
cuit is  joined  to  a  file  and  the  free  end  of  the  other  wire  is  drawn  over 
the  file  (Fig.  433),  the  circuit  is  rapidly  closed  and  broken,  producing 
a  shower  of  brilliant  sparks.  Without  the  electromagnet  in  the  cir- 
cuit, the  sparks  are  very  feeble. 

The  effect  of  the  coil  is  due  to  induction.  When  the  circuit  through 
the  coil  is  broken,  the  core  instantly  loses  its  magnetism,  and  a  direct 
E.M.F.  is  induced  in  the  coil,  just  as  if  a  strong  magnet  were  with- 
drawn from  it.  This  induced  E.M.F.  may  be  hundreds  of  times 
greater  than  that  of  the  battery  which  supplies  the  current.  Its  effect 
is  to  prolong  the  current  after  the  break  by  driving  it  across  the  gap, 
thus  producing  the 
spark.  With  even 
a  small  coil,  such  as 
that  of  a  small  elec- 
tric bell,  the  in- 
duced E.M.F.  at 
break  is  great 
enough  to  give  a 
shock,  if  the  bare 
wires  are  held  in 
the  moistened  fin- 
gers when  the  cir- 
cuit is  broken;  and 
with  a  large  coil  the 
shock  may  be  painfully  strong,  although  the  primary  current  is  sup- 
plied by  a  single  cell. 

Whenever  the  strength  of  the  current  in  a  coil  is  changing,  its 
magnetic  field  is  also  changing;  and  the  changing  field  reacts  induct- 


FIG.  433.  —  Self-induction. 


570  ELECTRODYNAMICS 

ively  on  the  coil,  just  as  it  does  on  a  secondary  coil  when  one  is  pres- 
ent. If  the  current  is  increasing,  as  at  the  instant  when  the  circuit 
is  closed,  the  induced  E.M.F.  is  inverse  and  opposes  the  primary  or 
battery  E.M.F.  Its  effect  is  to  retard  the  growth  of  the  current, 
which  therefore  requires  a  fraction  of  a  second  to  gain  its  full  value. 
When  the  circuit  is  broken  the  current  falls  to  zero  very  suddenly; 
and,  since  the  induced  E.M.F.  is  proportional  to  the  rate  of  change 
of  the  magnetic  field,  it  may  be  thousands  of  times  higher  at  "  break  " 
than  at  "  make." 

The  inductive  action  of  a  changing  current  on  itself  is  called  self- 
induction,  and  the  current  due  to  self-induction  at  break  is  called 
the  extra  current.  Self-induction  in  a  coil  increases  with  the  number 
of  turns,  and  is  enormously  increased  by  the  magnetic  action  of  a 
soft  iron  core.  A  coil  with  a  core,  placed  in  a  circuit  for  the  purpose 
of  producing  a  spark  at  break,  is  called  a  spark  coil.  Spark  coils 
are  used  in  battery  circuits  for  lighting  gas  jets,  and  for  igniting  the 
explosive  mixture  in  some  gas  and  gasoline  engines. 

466.  The  Induction  Coil.  —  The  induction  or  Ruhm- 
korff  coil  (Fig.  434)  is  an  instrument  for  generating  induced 

/?  G 


FIG.  434.  —  Induction  Coil. 

currents  of  very  high  potential.  A  simplified  diagram  of 
the  essential  parts  is  shown  in  Fig.  435.  These  are  an  iron 
core,  AB,  a  primary  coil  consisting  of  one  or  two  layers 
of  turns  of  large  insulated  wire,  a  secondary  coil  of  very 


ELECTROMAGNETIC  INDUCTION  571 

fine  wire,  well  insulated  and  often  many  miles  in  length, 
an  automatic  make-and-break  device  or  current  inter- 
rupter, CD,  which  is  included  in  the  primary  circuit,  and  a 
condenser,  E.  There  is  generally  also  a  device,  called  a 
switch,  for  reversing  the  current  through  the  primary  coil 
without  changing  the  battery  connections. 

When  a  battery  current  is  sent  through  the  primary  coil, 
it  magnetizes  the  iron  core,  and  the  core  attracts  the  iron 
block,  C,  which  is  supported  near  the  end  of  the  core  upon  a 
spring.  This  spring  is  the  movable  part  of  the  interrupter, 
and  the  primary  current  passes  between  it  and  the  point 
of  a  screw,  D,  against  which  it  rests.  By  the  attraction  of 
the  magnetized  core  the  spring  is  drawn  away  from  the 
point,  breaking  the  circuit.  The  core  instantly  loses  its  mag- 
netism, and  the  spring  flies  back  again,  closing  the  circuit. 
(This  action  is  the  same  as  in  the  electric  bell.)  The  pri- 
mary circuit  is  thus  closed 
and  broken  many  times 
every  second,  causing  alter- 
nately  an  inverse  and  a 
direct  induced  E.M.F.  in  the 
secondary  coil.  The  ends  l  1  }e' 

of  the  secondary  coil  are  FlG-  43S'~Diagc0rf1  of  an  Induction 
connected  with  the  binding 

posts,  R  and  G,  and  may  be  extended,  by  means  of  rods  or 
wires  attached  to  the  posts,  until  the  gap,  H,  is  made  as 
small  as  desired.  When  this  gap  is  not  too  great,  a  spark 
passes  between  the  terminals  with  every  interruption  of 
the  primary  current. 

The  maximum  length  of  the  spark  depends  upon  the 
induced  E.M.F.,  and  increases  with  the  number  of  turns 
in  the  secondary  coil  and  with  the  rate  of  change  of  the 
magnetic  field.  Since  this  change  is  much  more  abrupt 


572  ELECTRODYNAMICS 

at  break  than  at  make  (Art.  465),  the  spark  passes  only 
at  break.  The  purpose  of  the  condenser  (Art.  398)  is  to 
prevent  or  at  least  diminish  the  spark  in  the  primary  circuit 
at  the  interrupter,  and  thus  to  increase  the  abruptness  of 
the  break.  This  it  does  by  serving  as  a  temporary  reservoir 
into  which  the  extra  current  flows,  instead  of  jumping  across 
the  gap. 

The  induced  E.M.F.  of  a  coil  may  be  roughly  estimated  at  25,000 
volts  per  centimeter  of  the  longest  spark  that  it  will  give.  To  pro- 
duce a  spark  only  a  few  centimeters  in  length  requires  many  thou- 
sands of  turns  in  the  secondary.  The  sparking  distance  of  the  largest 
coils  runs  from  two  to  three  and  a  half  feet,  and  the  E.M.F.  is  from 
2,000,000  to  3,000,000  volts  or  even  higher.  For  large  coils  the  inter- 
rupter is  a  separate  mechanism,  different  from  that  described  above. 

The  induction  coil  has  many  important  uses.  The  so-called  physi- 
cian's battery  is  a  small  induction  coil,  operated  by  a  battery  current. 
The  handles  which  the  patient  holds  are  the  terminals  of  the  second- 
ary. Large  induction  coils  are  used  in  operating  X-ray  tubes  (Art. 
504)  and  in  wireless  telegraphy  (Art.  499) .  A  simple  form  of  induc- 
tion coil,  without  an  interrupter,  is  used  in  the  telephone  (Art.  483) ; 
and  another  form,  known  as  the  transformer,  is  an  indispensable 
factor  in  the  transmission  of  electrical  power  over  long  distances  (Art. 
479).  The  induction  coil  is  generally  preferred  to  the  primary  or 
spark  coil  for  igniting  the  charge  in  gas  and  gasoline  engines,  the  prin- 
cipal advantage  being  that  the  secondary  spark  will  jump  across  a 
gap  between  the  fixed  terminals  of  a  "spark  plug,"  whereas  the  pri- 
mary spark  requires  a  make-and-break  device  within  the  cylinder 
of  the  engine. 

467.  Current  Induction  in  the  Dynamo.  —  An  induced 
E.M.F.  can  be  generated  either  by  the  motion  of  a  magnetic 
field  within  a  stationary  coil,  or  by  the  motion  of  a  coil 
in  a  stationary  field.  The  latter  method  is  employed  in 
most  forms  of  dynamos.  The  small  dynamo  shown  in 
Fig.  436  is  an  example.  The  current  is  generated  in  a  set 
of  coils,  A,  which  are  wound  on  an  iron  core  to  increase 


ELECTROMAGNETIC  INDUCTION 


573 


FIG.  436.  —  Small  Bi-polar  Dynamo.  N,  S, 
Poles  of  Field  Magnet;  C,  Coil  of  Field 
Magnet;  A,  Armature;  P,  Driving  Pulley. 


the  induction.  The  coils  and  core  together  are  called  the 
armature.  The  armature  is  nearly  surrounded  by  the 
curved  poles,  N  and  5,  of 
an  electromagnet.  This 
is  termed  the  field  mag- 
net, since  it  produces  the 
magnetic  field  in  which 
the  armature  turns.  The 
coil ,  C,  of  the  field  magnet 
is  called  the  field  coil.  It 
is  connected  as  a  shunt 
to  the  external  circuit  (in 
this  dynamo),  and  takes 
a  small  part  of  the  cur- 
rent generated  in  the  armature.  The  armature  is  driven  at 
high  speed  by  a  belt  running  over  a  pulley,  P.  The  rota- 
tion of  the  armature  coils  in  the  field  of  the  electromagnet 
causes  the  induction. 

The  induction  in  each  coil  of  the  armature  runs  through 
a  complete  cycle  of  changes  during  each  revolution.  The 
nature  of  this  cycle  can  be  determined  with  the  aid  of  Fig. 
437,  which  represents  a  single  loop  of  an  armature  coil, 
AB,  turning  in  the  direction  of  the  curved  arrow,  between 
the  vertical  poles,  N  and  5,  of  the  field  magnet.  The  direc- 
tion of  the  lines  of  force  of  the  field  is  from  N  to  S,  or 
from  left  to  right  in  the  figure.  We  shall  suppose  that 
the  coil  starts  from  the  horizontal  position  and  makes  one 
complete  turn,  in  four  stages  of  90°  each.  At  the  start 
the  plane  of  the  coil  is  parallel  to  the  lines  of  force,  and 
hence  none  of  them  extend  through  it.  As  the  coil  turns 
into  the  vertical  position,  it  takes  in  a  constantly  increas- 
ing number  of  the  lines.  The  effect  is  the  same  as  if  the 
N  pole  of  a  magnet  were  thrust  into  the  coil  from  the  left 


574 


ELECTRODYNAMICS 


JT 


FIG.  437.  —  Simple  Alternator. 


side;  hence  the  direction  of  the  induced  E.M.F.  is  counter- 
clockwise, as  indicated  by  the  arrow-heads  (Art.  463).    As 

the  coil  turns 
from  the  vertical 
to  the  horizontal 
position  through 
the  second  90°, 
the  portion  of  the 
field  extending 
through  it  d e - 
creases  to  zero, 
and  the  induction 
is  the  same  as  if 
the  N  pole  of  a 
magnet  were 
withdrawn  from  the  coil.  Hence  the  direction  of  the  current 
is  clockwise.  As  the  coil  turns  from  the  horizontal  to  the 
vertical  position  again,  through  the  third  90°,  the  cross-sec- 
tion of  the  field  included  within  it  again  increases  to  a  maxi- 
mum, and  the  current  flows  counter-clockwise,  as  at  first. 
But  since  the  opposite  side  of  the  coil  now  faces  toward  the 
observer  (the  sides  A  and  B  being  interchanged),  the  direc- 
tion of  the  current  with  respect  to  the  coil  itself  is  really  the 
same  as  it  was  during  the  second  90°  of  the  revolution.  In 
other  words,  the  current  reverses  its  direction  in  the  coil  as 
the  coil  passes  the  vertical  (or  the  position  at  right  angles  to 
the  lines  of  force),  but  not  when  it  passes  the  horizontal  (or 
the  position  parallel  to  the  lines  of  force) .  Hence  a  second 
reversal  of  the  current  takes  place  as  the  coil  begins  the 
fourth  quarter  of  the  revolution,  i.e.  when  the  coil  passes 
the  vertical  with  side  A  below.  Thus  a  continuous  rotation 
of  the  coil  induces  an  alternating  current  in  it,  the  reversal 
of  the  current  taking  place  twice  during  each  revolution,  as 


ELECTROMAGNETIC  INDUCTION  575 

the  coil  passes  through  the  position  at  right  angles  to  the 
lines  of  force.  (If  the  poles  of  the  field  magnet  were  hori- 
zontal, and  the  lines  of  force  of  the  field  vertical,  as  in  Fig. 
436,  we  should  have  to  substitute  vertical  for  horizontal  and 
horizontal  for  vertical  throughout  the  above  discussion,  for 
the  induction  depends  upon  the  relative  position  and  motion 
of  the  coil  and  the  lines  of  force.) 

468.  The  Dynamo  Rule.  —  The  direction  of  the  induced  E.M.F. 
in  any  part  of  a  circuit,  as  the  sides  of  a  rotating  coil,  can  be  readily 
determined  by  the  dynamo  rule,  which  is  as  follows:  Extend  the 
thumb  and  the  first  and  second  fingers  of  the  right  hand  at  right 
angles  each  to  each;  then  turn  the  hand  so  that  the  first  finger  points 
in  the  direction  of  the  lines  of  force  of  the  inducing  field,  and  the  thumb 
in  the  direction  in  which  the  wire 
is  moving  across  the  lines  of  force. 
The  second  finger  will  then  point 
along  the  wire  in  the  direction  of 
the  induced  E.M.F.  (Fig.  438). 

Thus  for  the  side  A  of  the  coil 
in  Fig.  437  the  forefinger  points 
toward  the  right  (from  the  ./V" 
pole  to  the  5"  pole  of  the  mag- 
*net),  and  the  thumb  upward. 
The  second  finger  then  points  FlG"  438. -Right-hand  Rule  for  Deter- 
mining Direction  of  Induced  Current, 
in  the  direction  of  the  arrow- 
head, or  away  from  the  observer.  For  the  side  B  the  forefinger  points 
to  the  right  and  the  thumb  downward.  The  second  finger  then 
points  in  the  direction  of  the  arrow-head,  or  toward  the  observer. 

According  to  the  rule,  the  induced  E.M.F.  in  either  side  of  the  coil 
continues  in  one  direction  as  long  as  the  wire  continues  in  one  direc- 
tion (upward  or  downward)  across  the  lines  of  force,  and  reverses 
at  the  instant  when  the  wire  starts  across  the  lines  in  the  opposite  di- 
rection. Hence,  as  already  stated,  the  current  reverses  its  direction 
in  the  coil  as  the  coil  passes  through  the  position  at  right  angles  to 
the  lines  of  force.  The  induction  in  the  wire  at  the  ends  of  the  coil 
is  across  the  wire,  and  not  in  the  direction  of  its  length;  hence  it  has 
no  effect  on  the  current. 


576  ELECTRODYNAMICS 

469.  Arbitrary  Use  of  the  Term  "Line  of  Force."  Variation  of 
the  Induced  E.M.F.  in  a  Rotating  Coil.  —  A  line  of  force,  according  to 
our  previous  use  of  the  term  (Art.  375),  is  simply  a  mathematical  line 
indicating  the  direction  of  a  resultant  force.  It  has  no  real  existence. 
In  this  sense  a  line  of  force  passes  through  every  point  in  a  magnetic 
field;  hence  their  number  is  umlimited  or  infinite.  But  it  is  custom- 
ary to  speak  of  lines  of  force  as  if  they  really  existed,  and  were  pres- 
ent in  limited  numbers  throughout  a  magnetic  field,  in  proportion 
to  the  intensity.  It  is  arbitrarily  assumed  that,  in  a  field  of  unit 
intensity,  there  is  one  line  of  force  per  square  centimeter  of  the  cross- 
section  perpendicular  to  the  lines,  that  in  a  field  of  twice  this  inten- 
sity there  are  two  lines  per  square  centimeter,  etc.  This  is  only  a 
mathematical  fiction  ;  but  it  serves  a  useful  purpose  as  a  basis  for  stat- 
ing electromagnetic  relations  in  simple  and  definite  terms. 

In  the  first  place,  the  number  of  lines  of  force  per  sq.  cm.  of  the 
plane  perpendicular  to  the  lines  becomes  the  measure  of  the  intensity 
of  the  field  (Fig.  439) ;  and  hence  the  induction  in  any  circuit  is  pro- 
portional to  the  rate  of  change  in  the 
number  of  lines  of  force  passing 
through  it.  The  E.M.F.  induced  in 
any  part  of  a  circuit,  as  a  certain 
length  of  wire,  is  proportional  to  the 
number  of  lines  of  force  which  it 
crosses  or  "cuts"  in  a  second. 

An  armature  coil,   turning  at    a 

FIG.  439.  —  Strength  of  Magnetic    uniform  rate,  cuts  the  lines  of  force 
Field  is  Measured  by  Number    of  the   field  most   rapidly  when   its 


of   Lines  of  Force  per    Square 
Centimeter  of  Cross-section. 


sides  are  moving  at  right  angles  to 
the  lines,  e.g.  when  the  coil  AB  (Fig. 
437)  is  passing  through  the  horizontal  position.  The  induced  E.M.F. 
is  then  at  a  maximum.  As  the  coil  turns  through  90°  from  this 
position,  its  sides  cut  the  lines  of  force  more  and  more  obliquely,  and 
hence  at  a  diminishing  rate;  and  the  induced  E.M.F.  decreases  in 
proportion.  As  the  coil  passes  the  perpendicular  to  the  lines  of 
force,  its  sides  are  moving  parallel  to  the  lines  and  are  cutting 
none;  hence  the  induced  E.M.F.  is  then  zero.  In  one  revolution, 
starting  with  the  coil  perpendicular  to  the  lines,  the  induced  E.M.F. 
rises  from  zero  to  a  maximum,  decreases  to  zero,  rises  to  a  maximum 
in  the  opposite  direction,  and  again  decreases  to  zero. 


ELECTROMAGNETIC   INDUCTION 


577 


470.  Devices  for  Leading  the  Current  through  an  Ex- 
ternal Circuit.  —  The  alternating  current  generated  in  an 
armature  coil  may  be  led  off  through  an  external  circuit 
either  as  an  alternating  current  or  as  a  direct  current, 
depending  upon  the  mechanism  by  which  the  coil  is 


FIG.    440.  —  Side    and    End    View    of    Single    Loop    and 
Commutator. 

connected  with  the  terminals  of  the  dynamo.  In  the 
alternating-current  dynamo,  or  alternator,  this  mechanism 
consists  of  two  copper  collecting  rings,  c  and  d  (Fig.  437), 
mounted  on  the  armature  shaft.  They  are  insulated  from 
the  shaft  and  from  each  other,  and  an  end  of  the  coil  is 
connected  with  each.  The  rings  are  connected  with  the 


FIG.  441.  —  Direct-current  Dynamo  with  Single- loop  Armature. 

external  circuit,  HG,  by  means  of  stationary  terminals 
or  brushes,  e  and  /,  consisting  of  copper  strips  or  blocks  of 
carbon.  As  the  rings  turn  with  the  shaft,  the  brushes 
make  a  sliding  contact  with  them.  The  current  is  of  the 
same  character  in  the  external  circuit  as  in  the  coil. 


578 


ELECTRODYNAMICS 


In  the  direct-current  dynamo  a  device  known  as  a  com- 
mutator takes  the  place  of  the  collecting  rings.  For  a 
single-coil  armature  this  consists  of  a  copper  ring  split 
in  halves,  c  and  d  (Figs.  440  and  441),  with  the  parts  insu- 
lated from  each  other.  Fig.  440  shows  both  a  side  and  an 
end  view  of  the  coil  and  the  commutator.  In  the  other 
figure  are  end  views,  showing  the  position  of  the  brushes, 
e  and  /.  The  curved  arrow  indicates  the  direction  of  rota- 


FIG.  442.  —  Direct-current  Dynamo  with  Ring 
Armature. 

tion,  and  the  other  arrows  indicate  the  direction  of  the 
current.  The  small  cross  within  a  small  circle  represents 
the  tail  of  a  receding  arrow,  and  indicates  that  the  current 
in  that  side  of  the  coil  flows  from  the  observer.  The  dot 
within  a  small  circle  represents  the  head  of  an  approaching 
arrow. 

The  brushes  must  be  set  on  opposite  sides  of  the  commu- 
tator, in  such  positions  that  each  changes  contact  from  one 


ELECTROMAGNETIC    INDUCTION  579 

commutator  segment  to  the  other  at  the  instant  when  the 
current  changes  its  direction  in  the  coil.  Thus  each  seg- 
ment is  in  contact  with  brush/  while  it  is  positive,  and  with 
brush  e  while  it  is  negative.  The  current  in  the  external 
circuit  is  therefore  always  in  one  direction ;  but,  if  generated 
in  a  single  coil,  as  in  the  illustration,  it  is  pulsating  or  inter- 
mittent, rising  to  a  maximum  and  falling  to  zero  twice 
during  each  revolution  of  the  coil. 

471.  The  Direct-Current  Dynamo. —The  E.M.F.  in- 
duced in  the  armature  of-  a  dynamo  is  proportional  jointly 
to  the  total  number  of 
turns  in  all  the  coils,  to  -.--.— 
the  strength  of  the  field,  ~-j-\~- 
and  to  the  rate  of  rota-  ::i::_ 
tion.  The  number  of  -_-_-_"_ 
turns  varies  with  the  Irrrl- 
size  of  the  machine,  and  :_:  5 
with  the  desired  voltage. 

For  a  Small    Current  at       FlG-  443-  — Lines  of  Force  through  a  Ring 
.  Armature. 

high  voltage  the  arma- 
ture is  wound  with  many  turns  of  small  wire;  for  a  large 
current  at  a  low  voltage  it  is  wound  with  fewer  turns  of 
large  wire.  In  any  case  the  winding  is  evenly  distributed 
round  the  coil,  in  order  that  the  induction  may  be  con- 
stant. This  produces  a  steady  direct  current,  instead  of  a 
pulsating  one. 

Armatures  are  of  two  principal  types,  known  as  ring 
armatures  and  drum  armatures.  In  the  former  the  iron 
core  is  a  ring  (Fig.  442) ;  in  the  latter  it  is  a  cylinder  (Figs. 
436,  444,  and  445). 

The  winding  of  an  eight-coil  ring  armature  is  plainly  shown  in  Fig. 
442.  The  coils  are  all  wound  in  the  same  direction  and  are  joined 


580 


ELECTRODYNAMICS 


FIG.  444.  —  Winding  of  a  Four- 
coil  Drum  Armature. 


in  series.     Each  junction  between  coils  is  connected  with  one  of  the 

commutator  segments,  a,  b,  c,  etc.     The  lines  of  force  of  the  field 

crowd  into  the  iron  ring,  but  do  not  penetrate  the  space  within  it 

(Fig.  443) ;  hence  they  are  cut  only  by  the  outer  side  of  the  coils,  and 

there  is  no  induction  on  the  inside. 
By  applying  the  dynamo  rule  (Art. 
468)  it  will  be  found  that  the  direction 
of  the  induced  E.M.F.  is  as  shown  by 
the  arrow-heads.  The  current  thus 
flows  toward  the  positive  brush  BI  in 
both  the  right  and  the  left  halves  of  the 
armature.  As  each  coil  crosses  from 
the  left  to  the  right  side  above  or  from 
the  right  to  the  left  side  below,  the  in- 
duction in  it  falls  to  zero  and  reverses. 
Hence  the  induced  E.M.F.  is  always  in 

the  right  direction;  and,  as  there  are  always  three  active  coils  on 

each  side,  it  is  practically  constant. 

The  coils  of  a  drum  armature  surround  the  entire  core  (Fig.  444), 

and  induction  takes  place  on  both  sides,  as  explained  in  connection 

with  Fig.  437.     The 

coils   are  joined   in 

series  and  connected 

with    the    armature 

segments,  as  in  the 

ring  armature.    The 

plan   of  a   four-coil 

armature  is  shown  in 

Fig.  444.  The  arma- 
tures of  commercial 

dynamos     usually 

have  from  thirty  to 

one  hundred  coils. 
The  dynamos  thus 

far    considered    are 

bipolar,  i.e.  they  have  one  field  magnet  with  two  poles.     Four-pole 

dynamos  are  more  common  (Figs.  445  and  446).     In  these  the  poles 

are  alternately  N  and  5,  and  the  current  reverses  in  each  conductor 

as  it  passes  from  one  pole  to  the  next,  or  four  times  in-each  revolution. 


FIG.  445.  —  Parts  of  a  Four-pole,  Direct-current 
Dynamo  or  Motor. 


ELECTROMAGNETIC   INDUCTION 


If  an  armature  core  were  made  of  a  single  piece  of  iron,  currents 
would  be  induced  in  it  as  in  the  coils.  Such  currents  are  worse  than 
useless,  for  energy  is  wasted 
in  generating  them,  and,  be- 
sides, this  energy  is  trans- 
formed into  heat  in  the  core. 
If  this  were  permitted,  the 
armature  would  become  so 
hot  as  to  injure  or  destroy 
the  insulation.  Armatures 
are  therefore  built  up  of 
thin  disks  of  sheet  iron,  in- 
sulated from  one  another. 
These  disks  or  lamina 
extend  at  right  angles  to 
the  induced  E.M.F.; 
hence  the  current  tends  to 
pass  from  one  to  the  other,  but  is  prevented  by  the  insulation. 
Cores  of  this  description  are  called  laminated  cores. 


FIG.  446.  —  Dynamo  of  Fig.  445  Assembled. 


472.  Winding  of  the  Field  Coils.  —  A  direct-current 
dynamo  supplies  the  current  by  which  its  field  magnets  are 
excited.  In  one  type  of  machine  the  field  coils  consist  of 
many  turns  of  small  wire,  connected  as  a  shunt  to  the 


FIG.   447. — Shunt- 
wound  Dynamo. 


FIG.  448.  —  Series- 
wound  Dynamo. 


FIG.  449.  —  Com- 
pou nd  -  wound 
Dynamo. 


external  circuit  (Fig.  447).     These  are  called  shunt-wound 
dynamos.     In  the  series-wound  dynamo  (Fig.  448)   the 


582  ELECTRODYNAMICS 

field  magnets  are  wound  with  a  few  turns  of  large  wire 
connected  in  series  with  the  external  circuit,  and  the  entire 
current  flows  through  them.  In  the  compound- wound 
dynamo  (Fig.  449)  each  field  magnet  has  both  a  series  and 
a  shunt  coil. 

Each  variety  of  winding  has  certain  advantages  depending  on  con- 
ditions of  use.  Stated  briefly,  a  shunt-dynamo,  when  provided 
with  a  suitable  regulating  device,  is  a  constant-potential  machine.  It 
supplies  a  varying  current  at  a  constant  potential  on  a  circuit  of 
varying  resistance.  A  series  dynamo,  also  provided  with  a  regulating 
device,  is  a  constant-current  machine.  It  supplies  a  constant  current 
at  a  potential  which  varies  with  the  resistance  of  the  circuit.  Such 
machines  are  used  for  lighting  arc  lamps  in  series.  Compound  dyna- 
mos are  used  on  constant-potential  circuits  where  the  current  is  very 
fluctuating,  as  in  incandescent  electric  lighting  and  electric  street- 
car service. 

In  starting  a  dynamo,  the  current  is  at  first  very  weak,  since  the 
field  magnets  retain  very  little  magnetism;  but  this  small  current 
flowing  through  the  coils  of  the  magnets  strengthens  them,  producing 
a  stronger  current.  This  mutual  action  continues  until  the  magnets 
gain  their  full  strength. 

473.   Transformation  of  Energy  in  the  Dynamo.  —  The 

energy  of  the  current  generated  by  a  dynamo  is  derived 
from  the  mechanical  energy  expended  in  driving  the  arma- 
ture. A  part  of  the  energy  supplied  to  a  dynamo  is  lost 
in  overcoming  frictional  resistance,  and  there  is  a  further 
loss  in  the  coils  of  the  armature  and  the  field  magnets, 
owing  to  their  electrical  resistance.  These  losses,  taken 
together,  vary  from  about  15%  in  the  smaller  machines  to 
5%  in  the  larger  sizes.  Hence  the  efficiency  of  a  dynamo, 
as  a  device  for  converting  mechanical  energy  into  avail- 
able electrical  energy,  is  from  85  to  95%.  This  means  that 
the  power  required  to  run  the  armature  of  a  dynamo  at 
a  given  speed  is  from  seven  to  twenty  times  greater  while 


ELECTROMAGNETIC   INDUCTION  583 

the  machine  is  generating  a  current  than  it  is  when  the 
circuit  is  open. 

If  the  student  will  open  and  close  the  circuit  of  a  small  hand-power 
dynamo  while  he  is  running  it,  he  will  learn  by  personal  experience 
that  the  armature  carries  a  "load"  while  generating  a  current. 
The  sound  of  the  machine  tells  the  story  to  all  who  are  within 
hearing.  With  the  circuit  open,  the  moving  parts  emit  a  light, 
chattering  sound;  but  on  closed  circuit  the  sound  is  deep  and 
labored. 

The  added  resistance  to  the  rotation  of  the  armature  when 
it  is  generating  a  current  is  due  to  opposing  magnetic  forces, 
developed  according  to  Lenz's  law  (Art.  462).  The  in- 
duced current,  by  its  magnetic  action,  opposes  the  motion 
which  produces  it.  How  this  comes  about  can  be  under- 
stood by  referring  to  Fig.  437.  With  the  current  in  the 
direction  indicated,  the  N  side  of  the  coil  faces  the  N  pole 
of  the  field  magnet,  toward  which  it  is  turning.  The  motion 
of  the  coil  is  therefore  opposed  by  the  repulsion  of  the  N 
pole  and  also  by  the  attraction  of  the  5  pole  of  the  mag- 
net. In  fact  the  magnetic  forces  tend  to  turn  the  coil 
the  other  way  about.  These  opposing  forces  are  further 
increased  by  the  magnetized  core  of  the  armature.  This 
can  be  better  shown  from  Fig.  442.  The  current  in  the 
armature  coils  constantly  magnetizes  the  iron  core  in  right 
and  left  halves,  with  the  south  poles  of  the  two  semicir- 
cular magnets  together  at  the  top  and  their  north  poles 
together  at  the  bottom.  These  poles  are  each  constantly 
turning  toward  (but  never  reaching)  the  field  pole  which 
repels  it  and  away  from  the  pole  which  attracts  it. 

474.  The  Direct-Current  Motor.  —  The  energy  trans- 
formation in  a  dynamo  is  reversible.  When  a  current  is 
passed  through  the  armature  and  field  coils  of  a  dynamo 


ELECTRODYNAMICS 

the  armature  revolves  and  is  capable  of  driving  machinery. 
The  dynamo  then  becomes  an  electric  motor,  and  converts 
electrical  into  mechanical  energy.  "Manufacturers  sell 
their  standard  direct-current  dynamos  to  be  used  either 
as  generators  or  motors.  It  is  only  when  the  machines 
are  built  to  be  used  for  some  special  purpose  that  they 
can  not  be  conveniently  interchanged  in  their  action." 

As  shown  above,  the  magnetic  forces  acting  on  the  arma- 
ture of  a  dynamo  tend  to  turn  it  in  the  direction  opposite 
to  that  in  which  it  is  driven.  Hence  if  a  dynamo  is  sup- 

G  M 


FIG.  450.  —  Relation  of  Dynamo  to  Motor. 

plied  with  a  current  which  flows  through  its  field  and  arma- 
ture coils  in  thte  same  direction  as  the  current  which  the 
dynamo  itself  generates,  the  direction  of  rotation  will  be 
reversed ;  but  if  the  current  supplied  flows  in  the  same  direc- 
tion through  the  armature  coils  and  in  the  opposite  direc- 
tion through  the  field  coils,  the  rotation  will  be  in  the  same 
direction.  The  latter  case  is  shown  in  Fig.  450,  in  which 
G  represents  a  dynamo  supplying  a  current  to  a  motor,  M. 
Both  are  shunt-wound  machines  of  identical  construction. 
The  current  flows  in  the  same  direction  in  the  two  arma- 
tures; and  this  direction  is  such  as  to  maintain  a  double  5 


t 


ELECTROMAGNETIC    INDUCTION  585 

pole  at  the  top  of  the  core  and  a  double  N  pole  at  the  bot- 
tom. In  the  dynamo  these  poles  are  driven  in  opposition 
to  the  magnetic  attractions  and  repulsions  of  the  field  poles; 
in  the  motor  the  armature  is  turned  by  the  attractions  and 
repulsions  of  the  field  poles.  Since,  in  the  figure,  the  N 
field  pole  of  the  motor  is  in  the  position  of  the  5  field  pole 
of  the  dynamo,  and  vice  versa,  the  armatures  turn  in  the 
same  direction.  It  should  be  noted  that  the  current  leaves 
the  dynamo  armature  by  the  positive  brush  and  the  motor 
armature  by  the  negative  brush. 

475.  Transformation  of  Energy  in  the  Motor.  —  An 
E.M.F.  is  induced  in  the  armature  coils  of  a  motor,  when  it 
is  running,  for  the  coils  cut  the  lines  of  force  of  the  field 
just  as  they  do  in  the  dynamo.  By  applying  the  dynamo 
rule  to  the  motor  diagram  of  Fig.  450  it  will  be  found  that 
this  induced  E.M.F.  opposes  the  current  which  runs  the 
motor.  Hence  it  is  called  a  counter  or  back  E.M.F. 

This  action  can  be  shown  with  any  small  motor,  driven  by  a  bat- 
tery current  (Fig.  451).  An  ammeter,  A,  placed  in  the  circuit  will 
show  that  the  current  is  much  smaller  when  the  motor  is  running 
than  it  is  when  the  armature  is  held  at  rest.  When  the  motor  is 
run  at  different  speeds  by  varying  the  friction  at  the  pulley,  the  cur- 
rent decreases  as  the  speed  increases.  At  the  same  time  a  voltmeter, 
V,  will  show  that  the  fall  of  potential  in  the 
armature  increases  as  the  speed  increases. 
When  the  armature  is  at  rest,  the  fall  of 
potential  in  it  is  due  simply  to  the  resistance 
of  the  coils.  The  added  fall  of  potential 
when  the  motor  is  running  is  really  the  in- 
duced E.M.F.,  which  is  working  against  the 
current.  Hence  the  current  is  reduced,  as  FlG 

shown  by  the  ammeter.     It  is  as  if  a  smaller 

battery  were  placed  in  the  circuit  at  this  point,  with  its  E.M.F. 
opposed  to  that  of  the  principal  battery. 


586  ELECTRODYNAMICS 

If  the  armature  of  a  motor  is  not  permitted  to  turn,  the 
energy  expended  in  it  is  all  converted  into  heat  in  over- 
coming the  resistance  of  the  coils,  as  in  any  other  conductor; 
but  when  a  motor  is  running,  the  greater  part  of  the  elec- 
trical energy  (generally  from  85  to  95%  of  it)  is  expended 
in  overcoming  the  counter  E.M.F.  in  the  armature.  It  is 
this  part  of  the  energy  which  is  transformed  into  mechan- 
ical energy  by  the  motor  in  doing  work;  and  the  rate  at 
which  the  work  is  done  is  measured  in  watts  by  the  product 
of  the  current  and  the  counter  E.M.F.,  in  agreement  with 
equation  10,  page  555. 

v 

476.  Starting  a  Motor.  —  Motors  for  industrial  use  are  operated 
on  constant-potential  circuits.  These  may  be  no- volt  or  2 20- volt 
lighting  circuits,  or  separate  power  circuits,  usually  at  500  volts. 
The  latter  is  the  usual  voltage  for  street-car  service.  The  armature 
resistance  of  motors  is  very  small,  usually  only  a  fraction  of  an  ohm ; 
hence  if  the  full  voltage  of  the  circuit  were  applied  to  a  motor  in  start- 
ing it,  there  would  be  a  suddden  rush  of  current,  amounting  to  several 
hundred  amperes,  and  the  armature  would  be  ruined  by  overheating. 
Hence  the  current  is  turned  on  gradually  through  a  starting  box, 
which  contains  a  number  of  resistance  coils  in  series  with  each  other 
and  with  the  armature.  As  the  motor  gains  speed,  these  resistance 
coils  are  cut  out,  one  after  the  other,  by  moving  the  lever  arm  of  the 
starting  box.  Finally  all  of  the  box  resistance  is  cut  out,  leaving  the 
armature  connected  directly  to  the  circuit;  for  the  back  E.M.F.  is 
sufficient  to  prevent  an  excessive  current  when  the  motor  is  running 
at  full  speed. 

The  use  of  a  starting  box  may  be  observed  on  any  electric  street 
car.  The  entire  regulating  mechanism  is  contained  in  a  large,  up- 
right iron  box,  and  is  called  a  controller.  In  addition  to  the  resist- 
ance coils  of  an  ordinary  starting  box,  the  controller  contains  a  set 
of  switches  by  which  the  two  motors  of  the  car  can  be  joined  either  in 
series  or  in  parallel,  and  by  which  also  the  field  coils  of  each  motor  can 
be  joined  either  in  series  or  in  parallel.  Occasionally  a  motorman 
will  cut  out  the  starting  resistances  too  quickly.  The  motors  are 
protected  against  this  mischance  by  an  automatic  circuit  breaker 


ELECTROMAGNETIC   INDUCTION 


587 


which  breaks  the  circuit  by  the  action  of  an  electromagnet  when  the 
current  is  greater  than  it  should  be.  This  device  is  placed  under  the 
roof  of  the  car,  near  the  motorman.  Its  action  causes  a  loud  noise 
and  a  spark,  and  is  sometimes  the  occasion  of  needless  alarm  to  pas- 
sengers unacquainted  with  such  matters. 

477.  The  Alternating-current  Dynamo.  —  For  reasons  presented 
in  the  following  articles,  it  is  possible  to  transmit  electrical  energy 
for  power  purposes  over  distances  of  100  to  200  mi.,  by  means  of 
alternating  currents,  while  with  direct  currents  only  short  distances 
are  practicable.  The  alternating-current  dynamo,  or  alternator,  is, 
therefore,  of  very  great  industrial  importance.  The  study  of  alter- 
nating currents  and  alternating  current  machinery  is  very  extensive 
and  can  only  be  touched  upon  in  an  elementary  course  in  general 
physics. 

A  large  alternator  (Figs.  452  and  453)  has  many  field  poles,  always 
an  even  number.  They  are  alternately  N  and  S,  the  coils  of  all  the 
N  poles  being  wound  in  one  direction  and  those  of  the  5  poles  in  the 
opposite  direction. 
The  exciting  cur- 
rent is  direct,  and 
may  be  supplied 
from  a  direct-cur- 
rent winding  on 
the  armature  of  the 
alternator  itself,  or 
by  a  separate  ma- 
chine. The  arma- 
ture is  wound  with 
as  many  coils  as 
the  number  of  field 
poles,  and  alter- 
nate coils  are  oppo- 
sitely wound,  as 
shown  in  the  fig-^ 
ure.  As  the  arma- 
ture  turns,  the 

induced  E.M.F.  is  in  the  same  direction  round  all  the  coils  which  are 
passing  N  field  poles,  and  in  the  opposite  direction  round  all  the  coils 


FIG.  452.  —  Diagram  of  Multipolar  Alternator. 


588  ELECTRODYNAMICS 

which  are  passing  5  field  poles.  But,  owing  to  the  opposite  winding 
of  three  two  sets  of  coils,  the  induced  E.M.F.'s  are  all  in  the  same 
direction  through  the  armature  circuit  (i.e.  through  the  armature 
from  K2  to  Ki,  or  from  Ki  to  X2).  The  ends  of  this  circuit  are 
joined  to  collecting  rings,  as  shown;  and  the  current  passes  off  to 
the  external  circuit  by  one  and  returns  by  the  other.  The  current 
reverses  in  all  the  coils  at  the  same  instant,  as  they  turn  from  one 
field  pole  to  the  next;  and  the  current  reverses  in  the  external 
circuit  with  each  reversal  in  the  armature.  With  an  eight-pole 
machine  driven  at  the  rate  of  15  revolutions  per  second,  the  num- 
ber of  alternations  per  second  will  be  8  X  15  =  120.  Since  the 
current  runs  through  a  complete  series  of  changes,  or  one  cycle, 
during  the  interval  between  one  reversal  and  the  second  one  follow- 
ing, there  will  be,  in  the  present  instance,  60  cycles  or  current  waves 
per  second. 

Alternating  currents  for  electric  lighting  are  usually  6o-cycle  cur- 
rents. For  general  power  purposes  the  frequency  is  sometimes  as 
low  as  25  cycles  per  second.  Fig.  453  represents  a  modern  alter- 
nator, direct-connected  to  the  engine  which  runs  it.  The  term 
"direct-connected"  means  that  the  armature  of  the  dynamo  is 
mounted  on  the  shaft  of  the  engine.  Another  method  is  to  connect 
the  dynamo  and  engine  by  means  of  a  belt.  A  generating  unit 
includes  both  the  dynamo  and  its  engine  or  other  source  of  power. 
The  combination  of  a  dynamo  and  a  turbine  water-wheel  is  termed 
a  hydro-electric  unit.  Generating  units  of  5000  to  10,000  horse 
power  are  common  in  modern  electrical  power  stations. 

478.  Transmission  of  Electrical  Energy.  —  In  transmitting  elec- 
trical energy  over  a  line  for  use  at  a  distance,  a  certain  percentage  of 
it  is  lost  as  heat,  owing  to  the  resistance  of  the  conductor.  This 
loss  limits  the  distance  over  which  it  is  practicable  to  operate  power 
lines.  The  conditions  which  determine  the  loss  in  transmission  are 
disclosed  by  a  comparison  of  the  numerical  examples  presented  in 
the  table  below.  In  this  table  EI  denotes  the  E.M.F.  generated, 
E2  the  E.M.F.  at  the  end  of  the  line,  C  the  strength  of  the  current, 
and  R  the  resistance  of  the  line.  The  power  generated  is  EiC  watts 
and  the  power  delivered  for  use  EzC  watts.  The  fall  of  potential  in 
the  line  is  E\~E^  volts,  and  the  power  lost  in  transmission  (Ei—E2)C 
or  CzR  watts. 


ELECTROMAGNETIC    INDUCTION 


589 


Given:  — 

i 

2 

3 

4 

E.  M.  F.  delivered  (E2), 

500  volts 

500  volts 

1000  volts 

5000  volts 

Current  (C). 

10  amperes 

20  amperes 

10  amperes 

2  amperes 

Resistance  of  line  (R), 

25  ohms 

25  ohms 

25  ohms 

25  ohms 

Then:  — 

Power  delivered  (E2Q, 

5000  watts 

10,000  watts 

10,000  watts 

10,000  volts 

Power  lost  (C2R), 

2500  watts 

1  0,000  watts 

2500  watts 

100  watts 

Power  generated  (£]C), 

7500  watts 

20,000  watts 

1  2,  500  watts 

10,100  watts 

Fraction  of  power  lost, 

33-3% 

50% 

20% 

i  %  (nearly) 

E.  M.  F.  at  the  start  (£,), 

750  volts 

1000  volts 

1250  volts 

5050  volts 

It  will  be  seen  from  the  formulas  and  the  numerical  examples  that 
a  greater  amount  of  power  can  be  delivered  over  a  given  line  by  increas- 
ing either  the  current  or  the  voltage  at  which  it  is  delivered.  The 
loss  in  watts  is  determined  by  the  current  and  the  resistance  of  the 
line,  being  equal  to  C2  ft,  and  is  independent  of  the  voltage  (examples 
i  and  3  in  the  table).  Thus  if  the  power  generated  is  increased  ten- 
fold by  a  tenfold  increase  in  the  potential,  the  current  remaining  the 
same,  the  percentage  loss  in  transmission  is  reduced  nine  tenths. 

In  recent  years  rapid  progress  has  been  made  in  the  utilization  of 
water-power  through  the  agency  of  high-potential  currents,  generated 


FIG.  453.  —  Alternator  Direct-connected  to  Steam  Engine. 

in  electrical  power  stations  and  transmitted  to  distant  points  for  use 
in   manufacturing,    mining,    and    transportation.    The   voltage   at 


5QO  ELECTRODYNAMICS 

which  such  lines  operate  determines  the  distance  to  which  the  power 
can  be  transmitted  without  prohibitive  losses.  Higher  and  still 
higher  voltages  are  employed  from  year  to  year.  Fifteen  years  or 
so  ago  the  limit  was  30,000  volts.  At  the  present  time  various  lines 
are  operating  at  60,000  to  100,000  volts.  The  most  difficult  problem 
to  be  solved  in  this  development  has  been  to  provide  efficient  insula- 
tion. An  insulator  for  a  high  potential  line  is  made  of  the  best 
porcelain,  and  must  have  a  widely  ex- 
tended surface  thoroughly  protected  from 
moisture.  The  60,000- volt  insulator  shown 
in  Fig.  454  is  14  in.  in  diameter,  12  in. 
high,  weighs  26  lb.,  and  costs  about  $45. 
Such  an  insulator  is  required  at  each  point 
of  support  of  the  line  wire. 

One  of  the  largest  transmission  lines  in  the 
FIG  .  454.  —  Porcelain  Insu-   WOrld  is  the  Niagara-Syracuse-Auburn  line, 

lator  for  6o,ooo-volt  Circuit.        1-1,  •,  v 

which  transmits  30,000  horse-power  over  a 

distance  of  163  mi.  The  line  in  parts  is  designed  to  carry  60,000 
horse-power.  The  Colgate  plant,  Yuba  River,  California,  connects 
by  way  of  Oakland  and  Mission  San  Jose  to  a  line  222  mi.  in  length. 
This  plant  has  a  capacity  of  15,000  horse-power,  and  there  are  over 
100  sub-stations  on  1375  mi.  of  circuit  on  the  system.  At  McCalls 
Ferry  on  the  Susquehanna  River  a  dam  and  power  station  have 
recently  been  constructed  at  a  cost  of  nearly  $10,000,000.  The  power 
house  is  equipped  with  ten  twin-turbine  wheels,  each  with  a  capacity 
of  13,500  horse-power.  Similar  examples  of  power  development 
on  a  large  scale,  in  the  United  States  and  other  countries,  could 
be  named  by  the  score;  but  those  mentioned  will  serve  to  convey 
some  idea  of  the  tremendous  importance  which  modern  electrical 
science  has  given  to  nature's  perennial  source  of  energy,  —  run- 
ning water. 

479.  The  Transformer.  —  The  currents  generated  by  the  dyna- 
mos of  long-distance  power  lines  do  not  leave  the  power-house.  They 
are  alternating  currents,  generated  at  a  pressure  of  2000  to  6000  volts, 
and  are  sent  -through  the  primary  of  a  huge  induction  coil  of  special 
design,  called  a  transformer.  A  high-potential  current  is  induced 
in  the  secondary  coil  of  the  transformer;  and  this  is  the  current 
which  is  transmitted  over  the  line  to  sub-stations,  or  transformer 


ELECTROMAGNETIC    INDUCTION  591 

houses,  located  near  points  where  the  energy  is  to  be  used.  At  a 
sub-station  the  line  current  passes  through  the  high-potential  coil  of 
a  transformer,  inducing  a  current  at  a  relatively  low  voltage  in  the 
secondary  coil. 

The  essential  parts  of  a  transformer  are  an  iron  core,  and  two  coils 
having  an  unequal  number  of  turns.  A 
core  in  the  form  of  a  closed  loop  (a  ring 
or  a  rectangle)  is  more  efficient  than  a 
straight  one,  since  it  carries  all  the  mag- 
netic lines  of  force  through  both  coils 
(Fig.  455).  An  alternating  current  sent 
through  either  coil  magnetizes  the  core 
first  in  one  direction,  then  in  the  other;  FIG.  455.  —  Diagram  of  Trans- 
and  these  reversals  of  magnetism  induce 

an  alternating  E.M.F.  of  the  same  frequency  in  the  other  coil.  If 
this  coil  is  on  a  closed  circuit,  an  alternating  current  will  be  gener- 
ated in  it.  Disregarding  a  small 
percentage  of  loss  in  the  transfor- 
mation, the  E.M.F.'s  of  the  primary 
and  induced  currents  are  in  direct 
proportion  to  the  number  of  turns  in 
the  two  coils.  If  the  primary  current 
is  sent  through  the  coil  of  fewer 
turns,  the  transformation  will  be 
from  lower  to  higher  potential.  This 
is  the  action  of  the  step-up  trans- 
former used  at  generating  stations. 
Transformation  from  higher  to  lower 
potential  is  effected  by  sending  the 
primary  current  through  the  coil 
having  the  greater  number  of  turns. 
This  is  the  action  of  the  step-down 
transformer  used  at  sub-stations. 

A    first-class   commercial   trans- 
former (Fig.  456)  will  thus  transfer 
FIG.  456.  —  Commercial  Transformer    ,  o—       .    ,,         .     .   .     , 

Removed  from  its  Case.  from    93    to   98%    of    the   electrical 

energy  from  one  circuit  to  another 

completely  insulated  from  it.  If  Ei  and  Ci  denote  the  primary 
E.M.F.  and  current  respectively,  and  E2  and  C2  the  induced  E.M.F. 


592  ELECTRODYNAMICS 


and  current,  then  the  pawer  of  the  one  is  Eid  and  the  power  of 
the  other  E2C2.  Disregarding  the  small  loss  in  transformation, 
EiCi  =  E2C2  or  Ei:E2::C2:Ci;  i.e.  a  transformer  changes  the  E.M.F. 
and  the  current  strength  in  reciprocal  proportion.  Thus  if  the 
E.M.F.  is  increased  ten-fold,  the  induced  current  is  one-  tenth  as 
great  as  the  primary  current,  and  vice  versa. 

The  core  of  a  commercial  transformer  is  rectangular,  and  is  built 
up  of  thin  plates  of  soft  steel,  like  the  armature  core  of  a  dynamo. 
The  coil  on  each  side  contains  both  primary  and  secondary  windings. 

Low  Pressure  Mains     • 


Alternator 


imps 
Fig.  457.  —  Electric  Light  Circuit. 

the  one  surrounding  the  other  but  insulated  from  it.  The  winding 
of  fewer  turns  is  made  of  the  larger  wire,  since  it  carries  the  larger 
current.  Transformers  are  inclosed  in  iron  cases  for  protection. 

In  cities  where  alternating  currents  are  used  for  house  lighting, 
the  current  is  generated  at  a  relatively  high  pressure,  generally  either 
1 100  or  2200  volts.  This  current  is  distributed  over  high-pressure 
mains  to  convenient  points,  where  step-down  transformers  are  lo- 
cated (Figs.  457  and  458) ;  and  the  secondary  coils  of  the  transformers 
supply  the  current  for  the  lamp  circuits. 

480.  The  Magneto-Telephone. — The  method  of  using 
a  modern  telephone  is  a  very  simple  matter  indeed.  The 
receiver  is  placed  to  the  ear,  a  number  is  spoken  into  the 
transmitter,  and  in  a  moment  two  persons,  perhaps  many 
miles  apart,  are  talking  to  each  other  as  if  they  were  in 
the  same  room.  It  is  almost  as  simple  as  pushing  a  button 
to  "  turn  on  "  an  electric  light.  But  simplicity  of  use  in 
either  case  is  the  net  result  of  a  very  complex  and  wonder- 
ful application  of  scientific  principles.  A  first  glance  at 
the  intricate  mechanism  of  a  complete  telephone  system, 


ELECTROMAGNETIC    INDUCTION 


593 


including  the  subscriber's  telephone  and  the  central 
exchange,  where  any  one  of  several  thousand  subscribers 
can  be  connected  in  less 
than  ten  seconds  with  any 
other,  gives  the  impression 
that  no  one  but  an  expert 
could  make  anything  out  of 
it  all.  But  a  close  inspec- 
tion of  any  single  detail  will 
show  that  it  is  only  an  ap- 
plication of  some  one  or 
more  of  the  principles 
already  familiar  to  the  stu- 
dent. T 

book  will  permit  us  to  con- 
sider only  the  main  points. 

The  simplest  possible  electric  telephone  line  consists  of 
two  receivers,  permanently  joined  by  wires  (Fig.  459). 
Each  receiver  serves  also  as  a  transmitter.  This  is  the 
original  telephone  line,  invented  by  Alexander  Graham 
Bell  in  1876.  The  working  parts  of  each  instrument  are 
a  permanent  magnet,  M,  a  coil  of  fine  wire,  C,  and  a  disk  of 
thin  sheet  iron,  D.  The  disk  is  supported  all  round  its  edge, 
and  is  free  to  vibrate  like  the  head  of  a  drum  or  the  tym- 
panum of  the  ear.  When  the  speaker's  mouth  is  close  to 
the  disk  at  either  end  of  the  line,  the  disk  is  forced  to  vi- 
brate in*  unison  with 


r  *f  t  fV/o  FIG.  458.— Transformer  on  Electric  Light 
Pole.  H,  H,  high  potential  wires; 
L,  L,  low  potential  wires;  T,  trans- 
former. 


the  sound  waves 
which  beat  upon  it. 
Being  of  soft  iron,  it 
is  magnetized  more  or 
less  as  it  approaches  the  pole  of  the  magnet  or  recedes  from 
it  in  vibrating.  This  varies  the  strength  of  the  magnetic 


FIG.  459.  —  Diagram  of  Original  Telephone 
Line. 


594  ELECTRODYNAMICS 

field  within  the  coil  of  wire,  and  induces  a  current  in  it, 
first  in  one  direction,  then  in  the  other,  in  rapid  succession. 
This  current,  flowing  through  the  coil  of  the  other  tele- 
phone, alternately  increases  and  decreases  the  strength  of 
its  magnet.  When  the  magnet  is  strengthened,  it  draws 
the  disk  more  strongly;  when  it  is  weakened,  the  disk 
springs  back.  The  disk  of  the  receiving  telephone  thus 
repeats  the  movements  imparted  by  the  sound  waves  to 
the  disk  of  the  transmitting  telephone;  and,  by  its  vibra- 
tion, it  reproduces  the  sound  with  remarkable  accuracy. 
A  telephone  line  of  this  sort  does  not  require  a  battery. 
It  works  successfully  over  short  distances;  but  over  a  long 
line  the  current  is  too  weak  to  reproduce  intelligible  speech. 
The  Bell  telephone  has  continued  in  use  as  a  receiver;  but 
as  a  transmitter  it  soon  gave  place  to  a  device  based  on  an 
entirely  different  principle. 

The  parts  of  a  modern  receiver  are  shown  in  Fig.  460.    The  mag- 
net is  in  the  form  of  an  elongated  U,  in  order  that  both  poles  may  act 

on  the  disk.  A  short,  flat  bar  of  soft 
iron  is  fastened  to  each  pole,  and  about 
it  is  wound  a  coil  of  fine  wire.  These 
iron  cores  are  more  sensitive  to  a 
varying  current  in  the  coils  than 
permanently  magnetized  steel  would 
be.  The  coils  are  joined  to  the  line 
circuit  through  a  flexible  conducting 
cord,  which  carries  strands  of  small 
wire.  The  working  parts  are  inclosed 
in  a  hard  rubber  case,  the  cap  of 
which,  when  screwed  on,  holds  the 
disk  in  position.  The  telephone  re- 

FIG.  460. —  Modern  Telephone    ceiver    is  one  of  the  most  sensitive 
Receiver,  Dissected.  .  ,     ,       T  ,. 

instruments  ever  invented.  In  ordi- 
nary use  it  takes  only  one  ten-thousandth  of  an  ampere,  and  a  cur- 
rent one  thousand  times  smaller  than  this  produces  audible  sound. 


ELECTROMAGNETIC    INDUCTION  595 

481.  The  Microphone.  —  The  principle  of  the  telephone 
transmitter  is  beautifully  illustrated  by  the  simple  micro- 
phone, from  which  it  was 
developed.  This  instru- 
ment, as  its  name  implies, 
reproduces  faint  sounds 
with  increased  intensity. 
Its  action  depends  upon 
the  fact  that  the  electrical 
resistance  of  a  loose  contact  FIG.  461.  — Microphone  in 
between  two  conductors 
varies  with  the  pressure. 
A  contact  of  carbon  with  carbon  gives  the  best  results. 

A  common  form  of  microphone  is  shown  in  Fig.  461. 
The  pointed  ends  of  a  carbon  rod,  C,  rest  loosely  in  cavities 
in  carbon  supports,  A  and  B.  These  supports  are  fixed 
to  a  small  sounding  board,  and  are  joined  in  series  with  a 
Bell  receiver  and  a  battery  of  one  or  two  cells.  Vibrations 
of  the  sounding  board  are  transmitted  to  the  carbon  rods, 
causing  a  rapid  change  of  pressure  at  their  points  of  con- 
tact. A  slight  increase  of  pressure  enlarges  the  area  of 
contact  and  decreases  the  resistance.  This  permits  a 
larger  current  to  flow.  When  the  pressure  is  lessened, 
the  resistance  increases  and  the  current  is  reduced.  The 
fluctuating  current  varies  the  strength  of  the  receiver  mag- 
net, causing  the  disk  to  vibrate  in  unison  with  the  sounding 
board  but  with  a  greater  amplitude.  An  inaudible  rub- 
bing or  tapping  of  the  sounding  board  with  the  finger 
causes  the  receiver  to  emit  a  loud,  rattling  sound,  and  a 
watch  lying  on  the  board  is  heard  very  distinctly. 

The  microphone  was  invented  by  David  E.  Hughes,  an 
English  physicist,  in  1878.  Edison  made  a  similar  discov- 
ery of  the  action  of  loose  carbon  contacts  in  the  same  year; 


596  ELECTRODYNAMICS 

and  it  was  not  long  before  various  inventors  had  devised 
practical  telephone  transmitters  based  on  this  principle. 

482.  The  Granular-carbon  Transmitter.  —  Modern  transmitters 
are  of  the  granular-carbon  type.    The  details  of  construction  differ 

in  different  makes;  but  the  general  form 
shown  in  Fig.  462  is  typical.  M  is  the 
mouthpiece,  D  the  vibrating  diaphragm. 
The  latter  is  generally  of  aluminum,  and 
is  held  in  position  by  springs  not  shown 
in  the  figure.  C  is  a  small  metal  cup, 
covered  by  a  mica  diaphragm,  M'.  This 
flexible  cover  is  attached  to  the  principal 
diaphragm,  Z>,  by  a  short  screw,  S,  and 
vibrates  with  it.  The  battery  circuit 
connects  with  two  carbon  disks,  E  and  E'. 
E  is  attached  to  the  bottom  of  the  cup  and  is  stationary;  E'  vi- 
brates with  'the  mica  diaphragm.  The  space  between  the  disks 
is  loosely  filled  with  small  carbon  granules,  which  serve  to  conduct 
the  current  between  E  and  E1.  These  granules  are  subjected  to  a 
varying  pressure,  due  to  the  vibration  of  E';  and  as  there  are  many 
points  of  loose  contact,  the  variation  of  the  resistance  is  large. 
The  corresponding  fluctuations  of  the  battery  current  are  there- 
fore much  greater  than  the  feeble  currents  generated  in  the  Bell 
receiver,  when  used  as  a  transmitter. 

483.  A  Complete  Telephone  Line.  —  The  simplest  tele- 
phone line,  complete  in  itself,  is  one  which  is  used  only 
for   communication   between    two   points.     This   requires 
at  each  end  of  the  line  a  transmitter,  a  receiver,  an  electric 
call-bell,  a  battery,  and  switching  devices  for  making  the 
necessary  connections.     This  apparatus,  with   the  excep- 
tion of  the  battery,  is  all  assembled  in  the  battery-call 
telephone  (Fig.   463).     The  term  "battery-call"   signifies 
that  the  battery  supplies  the  current  for  ringing  the  bell 
as  well  as  for  talking.     (This  is  possible  only  on  short  lines. 
On  a  long  line  the  bell  requires  a  more  powerful  source  of 


ELECTROMAGNETIC    INDUCTION 


597 


current,  and   a   small   hand-power   dynamo   or  magneto- 
generator  is  used  for  this  purpose.) 

Fig.  464  is  a  diagram  of  the  connections  in  a  battery-call  tele- 
phone. When  the  receiver  is  on  the  hook,  its 
weight  pulls  the  hook  down,  bringing,  it  into 
electrical  contact  with  a  terminal  at  a.  This 
connects  the  bell  with  the  line,  for  the  purpose 
of  receiving  signals.  A  signal  is  sent  by  press- 
ing the  button,  B,  which  brings  the  spring,  K, 
into  contact  with  the  terminal,  g.  This  con- 
nects the  battery  with  the  line,  and  rings  the 
bell  at  the  other  station.  When  the  receiver 
is  taken  from  the  hook,  the  hook  is  pushed 
up  by  a  spring.  This  disconnects  the  bell  at 
a,  closes  the  local  battery  circuit  at  d,  and 
connects  the  receiver  with  the  line  at  h.  The  F  l  G  •  463.  —  Battery-call 
battery  circuit  includes  the  transmitter  and 

the  primary  winding,  P,  of  a  small  induction  coil.     The  secondary 
winding,  S,  is  included  in  the  line  circuit,  in  series  with  the  receiver. 


Line  or 


Ground 


FIG.  464.  —  Diagram  of  a  Battery-call  Telephone. 

The  induction  coil  serves  as  a  miniature  step-up  transformer.  It 
may  be  dispensed  with  on  short  lines;  but  it  is  an  advantage  if  not 
a  necessity  on  long  lines,  owing  to  the  greater  resistance  to  be 
overcome. 


598  ELECTRODYNAMICS 

484.  The    Telephone  Exchange.  —  The    individual    subscribers' 
lines  of  a  telephone  system  or  exchange  all  terminate  in  a  switchboard 
at  the  central  station.    This  switchboard  is  so  contrived  that  the  oper- 
ators can  connect  any  line  with  any  other  by  means  of  conducting 
cords.     At  each  end  of  a  cord  is  a  plug,  provided  with  metal  terminals. 
When  the  plug  is  inserted  in  a  small  hole,  about  the  size  of  a  lead  pen- 
cil, its  terminals  are  brought  in  contact  with  two  springs,  which  form 
the  terminals  of  the  subscriber's  line.     On  a  large  switchboard  as 
many  as  5000  to  10,000  such  terminals  are  within  the  reach  of  a 
single  operator. 

The  details  of  a  modern  switchboard  and  the  auxiliary  apparatus 
necessary  for  its  operation  are  numerous  and  complicated.  A  stor- 
age battery  at  the  central  station  supplies  the  " talking  current"  for 
all  lines  of  the  system,  and  a  dynamo  supplies  an  alternating  current 
for  ringing  the  bells.  A  subscriber  calls  "  central "  simply  by  remov- 
ing the  receiver  of  his  telephone  from  the  hook.  The  current  which 
then  flows  over  his  line  operates  a  relay  in  the  central  station.  This 
closes  a  local  circuit  through  a  tiny  electric  lamp,  mounted  in  the 
switchboard  beside  the  terminal  of  the  subscriber's  line.  The  oper- 
ator connects  her  telephone  with  the  line  indicated  by  the  lamp, 
learns  what  number  is  wanted,  rings  the  bell  on  that  line,  then  con- 
nects it  with  the  line  of  the  calling  subscriber.  The  lamp  is  auto- 
matically cut  out  when  connection  is  made  with  the  line;  and  when 
the  receiver  of  either  telephone  is  hung  up,  another  lamp  lights,  as 
a  signal  for  the  operator  to  disconnect. 

In  an  automatic  exchange  no  operators  are  required,  as  all  connec- 
tions are  made  by  automatic  devices,  operated  by  electromagnets. 

485.  Danger  from  Electric  Currents.  —  Electricity  has  come  to 
be  such  an  important  factor  in  daily  life  that  every  one  should  know 
its  real  dangers,  and  should  not  be  troubled  with  imaginary  ones. 
Telephone,  telegraph,  and  incandescent  lighting  currents  are  not  at 
all  dangerous.     At  the  worst  a  lighting  current  at  no  volts  will  give 
an  unpleasant  shock,  and  a  current  at  220  volts  a  severe  "jolt." 
The  exposed  metal  parts  of  sockets  and  lamps  are  insulated  from  the 
circuit;  hence  in  ordinary  use  there  is  no  opportunity  to  come  in  con- 
tact with  the  current.    The  circuits  for  street-car  lines  and  for  general 
power  purposes  are  commonly  operated  at  500  volts.     Contact  with 
bare  wires  at  this  voltage  is  distinctly  dangerous,  especially  in  the 


ELECTROMAGNETIC    INDUCTION  599 

case  of  alternating  currents,  but  seldom  fatal.  Such  currents,  how- 
ever, will  kill  a  horse.  The  distributing  mains  which  run  to  the  trans- 
formers on  incandescent  lighting  systems,  the  circuits  of  street  arc 
lamps,  and  long-distance  transmission  lines  in  general  carry  currents 
at  1000  volts  or  higher.  Contact  with  such  wires  means  death,  as 
a  rule;  and  linemen  who  have  occasion  to  climb  the  poles  on  which 
the  wires  are  strung  are  now  and  then  victims  of  this  mischance. 
It  sometimes  happens  that  a  wire  carrying  a  high-tension  current 
breaks  and  falls  to  the  ground,  or  comes  in  contact  with  telephone  or 
incandescent  light  wires.  Such  an  accident  is  a  serious  menace  to 
life  and  property;  but,  fortunately,  it  is  a  very  rare  occurrence. 

Every  electric  circuit  should  be  regarded  with  suspicion,  unless 
its  character  is  known.  A  wire  carrying  a  deadly  current  differs  in 
no  wise  in  appearance  from  one  which  is  perfectly  harmless. 

When  current  is  taken  from  a  lamp  socket  for  ironing  or  cooking, 
the  circuit  should  never  be  broken  by  turning  the  key  of  the  socket. 
A  special  plug  switch,  provided  for  this  purpose,  should  always  be 
used.  The  parts  of  a  lamp  switch  are  designed  to  break  only  a  small 
current,  such  as  is  taken  by  a  lamp.  A  larger  current  is  very  likely 
to  form  an  arc  and  burn  out  the  switch. 

If  in  any  emergency  it  is  necessary  to  handle  a  "live  wire"  at 
a  dangerous  or  disagreeable  voltage,  it  should  be  remembered  that 
insulation  protects.  A  few  thicknesses  of  dry  cloth  between  the  wire 
and  the  hand  renders  500  volts  harmless.  Again,  the  current  that 
will  pass  through  the  body  depends  upon  the  resistance  of  the  cir- 
cuit of  which  the  body  forms  a  part.  If  only  one  wire  of  the  circuit 
is  touched,  the  current  passes  through  the  body  to  the  ground.  If 
any  fairly  good  insulator  is  interposed  in  this  path,  as  when  the  person 
is  standing  on  a  dry  board,  the  current  is  very  small  compared  with 
what  it  would  be  if  he  were  standing  on  the  ground,  especially  when 
the  ground  is  damp.  Not  the  slightest  shock  is  felt  on  touching  one 
wire  of  a  2  20- volt  circuit,  provided  no  other  part  of  the  body  is  in 
contact  with  a  better  conductor  than  wood;  but  if  both  wires  are 
touched  at  the  same  time,  the  shock  is  rather  severe,  for  the  body 
then  receives  the  full  voltage. 

Knowing  the  danger  of  high-potential  currenfs,  it  is  often  a  matter 
of  surprise  to  students  that  an  electrostatic  machine  or  an  induc- 
tion coil  which  works  at  30,000  volts,  or  even  higher,  can  be  treated 
as  a  plaything  without  fear  of  serious  consequences.  The  reason  is 


600  ELECTRODYNAMICS 

that  these  machines  develop  a  high  potential  only  on  open  circuit. 
Touch  the  knobs  of  an  electrostatic  machine  and  the  potential  in- 
stantly falls  practically  to  zero  (Art.  405).  Touch  the  terminals  of 
a  small  induction  coil  and  the  potential  falls  to  a  very  moderate  and 
harmless  value.  On  the  other  hand,  the  voltage  of  a  lighting  or 
power  current  is  maintained  on  closed  circuit,  by  the  action  of  a 
powerful  dynamo. 

PROBLEMS 

1.  If  a  dynamo  is  run  by  a  motor,  could  the  current  generated  by  the 
dynamo  be  used  to  run  the  motor? 

2.  In  what  respects  is  a  D'Arsonval  galvanometer  like  a  motor?    In 
what  respects  does  it  differ? 

3.  How  would  you  reverse  the  direction  of  rotation  of  a  motor? 

4.  The  lamps  of  a  street  car  are  lighted  by  the  5oo-volt  current  taken 
from  the  trolley  wire.     How  must  loo-volt  lamps  be  connected  for  this 
purpose? 

5.  A  dynamo  generates  a  current  of  50  amperes,  at  a  pressure  of  500 
volts,  on  a  line  whose  resistance  is  2  ohms.     Find  (a)  the  power  generated, 
(b)  the  power  lost  in  the  line,  (c)  the  power  delivered,     (d)  What  per  cent 
of  the  power  is  lost  in  the  line? 

6.  What  per  cent  of  the  power  would  be  lost  in  the  above  line  if  the 
dynamo  generated  25  amperes  at  a  pressure  of  1000  volts? 

7.  A  dynamo  supplies  current  for  lighting  5000  no- volt  lamps,  each 
taking  .5  ampere.     Allowing  10%  loss  in  the  wires  and  the  transformers, 
what  power  in  kilowatts  must  the  dynamo  generate? 

8.  The  current  taken  from  a  trolley  wire  returns  to  the  power  house 
through  the  rails.     Why  is  it  impossible  to  get  a  shock  from  the  rails? 

9.  If  the  primary  coil  of  a  transformer  has  800  turns  and  the  second- 
ary  200   turns,  what   voltage   will   be  induced   in   the   secondary   by  a 
primary  voltage  of  220? 

X.   CHEMICAL  EFFECTS  OF  THE  ELECTRIC  CURRENT 

486.  Electrolys^.  —  If  energy  is  given  out  in  any  phys- 
ical or  chemical  process,  an  equal  amount  of  energy  is  taken 
in  when  the  process  is  reversed.  Thus,  as  we  have  seen, 
a  gram  of  steam  gives  out  537  calories  of  heat  in  condensing 


CHEMICAL  EFFECTS  OF   CURRENTS  601 

at  1 00°,  and  537  calories  must  be  supplied  to  a  gram  of 
water  at  100°  to  vaporize  it.  Similarly  water  gives  out 
heat  in  freezing  and  absorbs  an  equal  amount  of  heat  in 
melting.  Energy  is  put  into  the  spring  of  a  watch  in  wind- 
ing it,  and  is  paid  out  again  as  the  spring  unwinds  itself. 
The  solar  energy  utilized  by  a  plant  in  separating  the  car- 
bon from  the  carbon  dioxide  of  the  air  is  recovered  as  heat 
when,  in  decaying  or  burning,  the  carbon  of  the  plant  unites 
with  the  oxygen  of  the  air  again  (Art.  244). 

Since,  under  suitable  conditions,  various  chemical  changes 
produce  electrical  energy,  as  in  electric  cells  of  all  kinds,  it 
is  reasonable  to  suppose  that,  by  an  expenditure  of  electrical 
energy,  these  changes  could  be  reversed.  In  the  simple 
cell,  for  example,  the  current  is  generated  by  the  action 
of  dilute  sulphuric  acid  on  zinc,  the  product,  zinc  sulphate 
(ZnSO4),  remaining  in  solution  (Art.  407).  Reversing  the 
process,  metallic  zinc  is  obtained  from 
a  solution  of  zinc  sulphate  by  means 
of  an  electric  current,  as  in  the  follow- 
ing experiment.  A  bent  tube  (Fig. 
465)  is  partly  filled  with  the  solution. 
A  narrow  strip  of  platinum,  soldered 
to  a  wire,  is  placed  in  the  liquid  in 
one  arm  of  the  tube  and  connected  FIG.  4 6 5 .  —  Electrolytic 
with  the  positive  pole  of  a  battery  of 
three  dry  cells  in  series;  and  an  iron  or  a  copper  wire  is  in- 
serted in  the  other  arm  of  the  tube,  and  connected  with  the 
negative  pole  of  the  battery.  While  the  current  is  flowing, 
bubbles  continue  to  rise  from  the  platinum  terminal;  and, 
after  a  few  seconds,  it  will  be  found  that  the  iron  or  copper 
wire  is  covered  with  a  layer  of  zinc.  The  current  in  pass- 
ing through  the  solution  decomposes  the  zinc  sulphate,  and 
carries  the  zinc  with  it  to  the  negative  terminal.  The  other 


602  ELECTRODYNAMICS 

product  of  the  decomposition,  consisting  of  sulphur  and  oxy- 
gen in  combination  (SO4),  goes  in  the  opposite  direction  to 
the  positive  terminal.  Here  it  unites  with  hydrogen  from 
the  water,  forming  sulphuric  acid  and  setting  oxygen  free. 
This  oxygen  forms  the  bubbles  that  rise  from  the  strip  of 
platinum.  The  presence  of  the  sulphuric  acid  can  be  shown 
by  adding  a  few  drops  of  blue  litmus  solution  to  the  con- 
tents of  the  tube  before  the  current  is  passed;  for  the  acid 
changes  the  color  of  the  liquid  from  blue  to  red  about  the 
platinum  terminal. 

The  above  experiment  is  an  example  of  the  chemical 
decomposition  of  a  compound  by  means  of  an  electric  cur- 
rent sent  through  a  solution  of  the  substance.  The  process 
is  called  electrolysis  (electro-analysis),  the  substance  de- 
composed is  termed  an  electrolyte,  and  the  vessel  in 
which  the  process  is  carried  out  an  electrolytic  cell.  The 
terminal  by  which  the  current  enters  the  solution  is  called 
the  positive  electrode  or  anode;  the  one  by  which  it  leaves 
the  liquid  is  the  negative  electrode  or  cathode.  (These 
terms  are  from  the  Greek,  meaning,  ode,  way  or  path;  an, 
up;  cat,  down;  electrode,  a  way  for  electricity;  anode,  the 
way  up  or  against  the  current ;  cathode,  the  way  down  or  with 
the  current.) 

If  an  anode  of  zinc  is  substituted  for  the  platinum  in  the  experi- 
ment, the  sulphur-oxygen  product  of  the  decomposition  (864)  unites 
with  it,  forming  more  zinc  sulphate.  The  anode  thus  loses  as  much 
zinc  as  the  cathode  gains,  and  the  strength  of  the  solution  remains 
constant.  Similar  results  are  obtained  with  copper  electrodes  and 
a  solution  of  copper  sulphate.  The  anode  wastes  away  and  an  equal 
weight  of  copper  is  deposited  upon  the  cathode.  Water  can  be  elec- 
trolyzed  in  a  cell  containing  dilute  sulphuric  acid  and  platinum  elec- 
trodes. The  acid  takes  part  in  the  chemical  changes;  but  only  the 
water  is  consumed,  hydrogen  appearing  at  the  cathode  and  oxygen 
at  the  anode. 


CHEMICAL  EFFECTS  OF   CURRENTS 


603 


Cofhod* 


FIG.  466.  —  "Migration"  of  the  Ions  in  an 
Electrolytic  Cell. 


487.  Ionic  Theory  of  Electrolysis.  —  In  a  solution  of  zinc  sul- 
phate a  certain  percentage  of  the  molecules  are  dissociated,  forming 
positive  zinc  ions  (Zn++)  and  negative  sulphions  (SC>4~  ~)  (Art.  407). 
When  the  electrodes  of  a  battery  circuit  are  placed  in  the  solution 
(Fig.  466),  the  negatively  charged  cathode  attracts  the  positive  and 
repels  the  negative  ions,  while 
the  positively  charged  anode 
does  just  the  opposite.  The 
result  is  a  slow  drift  of  the 
positive  and  negative  ions  in 
opposite  directions  through 
the  solution.  With  a  battery 
E.M.F.  of  three  or  more  volts, 
the  zinc  ions,  on  arriving  at 
the  catKode,  give  up  their 
positive  charges  to  it,  and 
adhere  to  the  surface,  form- 
ing a  layer  of  metallic  zinc. 
With  a  platinum  anode,  the 
sulphions,  on  coming  in  con- 
tact with  it,  give  up  their  negative  charges,  and  immediately  decom- 
pose water  molecules,  forming  sulphuric  acid  and  setting  oxygen  free. 
When  a  zinc  anode  is  used,  the  sulphion  acts  on  it,  and  new  zinc 
ions  are  formed  as  rapidly  as  they  are  deposited  from  the  solution  at 
the  cathode. 

A  current  of  electricity  in  solid  conductors  and  in  molten  metals 
passes  through  matter;  in  solutions  of  acids,  bases,  and  salts  (electro- 
lytes) the  electricity  is  transported  by  matter,  in  the  form  of  positive 
and  negative  charges  of  moving  ions.  All  ions  of  one  kind  carry 
equal  charges;  and  the  charge  on  any  kind  of  ion  is  either  equal  to  the 
charge  on  a  hydrogen  ion  or  is  an  exact  multiple  of  it  The  charge  of 
the  hydrogen  ion  is  thus  the  natural  unit  of  electricity.  It  is  denoted 
by  a  single  plus  sign  in  ionic  symbols,  as  in  H+  for  the  hydrogen 
ion  and  Na+  for  the  sodium  ion.  Zinc  and  copper  ions  carry  double 
charges;  hence  their  symbols  are  Zn++  and  Cu++.  The  unit 
negative  charge  is  represented  by  a  single  minus  sign,  as  in  the 
symbol  Cl  ~  for  the  chlorine  ion  and  OH~  for  the  hydroxide  ion. 
The  sulphion  (SO4~~)  carries  a  double  negative  charge.  The  fol- 
lowing are  further  examples  of  common  electrolytes. 


604  ELECTRODYNAMICS 

Substance  Ions  formed  in  solution 

Sulphuric  acid,  H2SO4  ^  H+  +  H+  +  SO4~  ~ 

Zinc  sulphate,  ZnSO4  ^  Znf+  +  SO4~ 

Copper  sulphate,  CuSO4  ^  Cu+ +  +  SO4~ 

Hydrochloric  acid,  HC1  ^  H+  +  Cl~ 

Sodium  chloride,  NaCl  ^  Na+  +  Cl~ 

Zinc  chloride,  ZnCl2  ^  Zn++  +  Cl~  +  Cl' 

Silver  nitrate,  AgNO3  —  Ag+  +  NO3~ 

Sodium  hydroxide,  NaO  ^  Na+  +  OH~ 

Aluminum  oxide,  A12O3  ^  2A1+  +  +   -f  36" ' 

In  every  case,  the  sum  of  the  negative  charges  is  equal  to  the  sum 
of  the  positive  charges.  This  is  shown  experimentally  by  the  fact 
that  the  solution  as  a  whole  has  no  charge,  as  would  be  the  case  if 
there  were  an  excess  of  either  positive  or  negative  electricity  within  it. 

488.  Laws  of  Electrolysis.  —  Since  in  an  electrolyte  the  current 
is  carried  only  as  charges  of  the  ions  and  all  like  ions  carry  equal 
charges,  the  amount  of  any  one  substance  liberated  by  electrolysis  is  pro- 
portional to  the  quantity  of  electricity  (ampere-hours]  which  passes 
through  the  electrolyte.  Thus  it  is  found  by  experiment  that  a  current 
of  one  ampere,  flowing  for  one  hour  through  a  solution  of  any  silver 
salt,  e.g.  silver  nitrate  or  silver  cyanide,  always  deposits  4.025  g.  of 
silver.  Similarly  a  current  of  one  ampere  liberates  in  one  hour 
.0376  g.  of  hydrogen  from  any  acid  solution,  1.203  g-  of  zmc  from 
any  electrolyte  whose  positive  ions  are  zinc- ions,  2.444  g.  of  gold 
from  a  solution  of  a  gold  compound,  etc. 

Having  determined  the  quantity  of  any  substance  which  a  given 
current  deposits  from  solution  in  a  given  time,  the  problem  can  be 
reversed  and  the  strength  of  an  electric  current  determined  by  find- 
ing the  weight  of  the  substance  that  the  current  deposits  in  a  certain 
time.  Since  mass  and  time  can  be  measured  with  very  great  accu- 
racy, the  method  serves  as  the  basis  for  the  practical  definition  of  the 
ampere.  "The  international  ampere,  as  thus  defined,  is  the  steady 
current  which  deposits  silver  at  the  rate  of  .ooiuSg.  per  second 
(4.025  g.  per  hour)  from  a  solution  of  silver  nitrate  in  water,  the  solu- 
tion being  of  a  given  fixed  strength  to  insure  regular  action." 

The  quantities  of  different  substances  liberated  from  their  solutions  by 
the  passage  of  equal  quantities  of  electricity  are  chemically  equivalent  to 


CHEMICAL  EFFECTS  OF  CURRENTS  605 

one  another.  The  term  "chemically  equivalent  quantities"  means 
that  there  is  just  enough  of  the  one  to  displace  the  other  completely  in 
a  chemical  change.  Thus  4.025  g.  of  silver  will  displace  .0376  g.  of  hy- 
drogen from  nitric  acid  (HNOs)  in  forming  silver  nitrate (AgNOs). 

The  two  statements  above  in  italics  are  known  as  Faraday's  laws 
of  electrolysis.  They  express  a  quantitative  relation  between  elec- 
tricity and  matter,  which  is  explained  in  part  by  the  ionic  theory  of 
solutions  (Art.  487).  This  is  supplemented  by  the  new  theory  con- 
cerning the  nature  of  electricity  and  its  relation  to  matter  (Art.  512). 

489.   Industrial     Applications     of     Electrolysis.  —  The 

industrial  applications  of  electrolysis  cover  a  wide  range, 
and  are  increasing  in  number  and  importance  from  year 
to  year.  Electrolytic  processes  are  of  two  general  types. 
In  electroplating  and  electro  typing  the  current  deposits  a 
thin  film  of  metal  —  gold,  silver,  nickel,  or  copper  —  upon 
a  prepared  surface.  In  electrometallurgy  the  current  per- 
forms the  work  of  chemical  decomposition  by  which  cer- 
tain of  the  metals  are  obtained  from  the  ores  and  minerals 
in  which  they  occur. 

Electrotyping  is  a  process  of  reproducing  pages  of  type  and  wood- 
cuts and  other  illustrations  by  an  electroplating  of  copper.  The 
matter  to  be  printed  is  first  set  up  in  common  type,  and  a  mold  of 
this  is  made  in  wax  by  pressing  it  hard  upon  the  type.  The  impressed 
side  of  the  mold  is  very  thinly  covered  with  powdered  graphite  to 
make  it  a  conductor  of  electricity.  The  mold  is  then  suspended  as 
the  cathode  in  an  acid  solution  of  copper  sulphate;  the  anode  is  a 
plate  of  copper.  When  the  current  is  passed,  copper  is  dissolved 
from  the  anode  and  deposited  as  a  thin  sheet  upon  the  mold,  forming 
an  exact  copy  of  the  original.  This  sheet  is  removed  from  the  mold, 
"backed  up"  by  a  filling  of  type  metal  to  give  it  strength,  and 
mounted  on  a  wooden  block.  It  is  then  ready  for  use  in  printing. 
Most  books,  are  now  printed  from  electrotype  plates,  which,  as  a 
rule,  are  preserved  for  many  years,  and  may  be  used  again  and 
again  in  printing  new  editions. 

In  electroplating,  the  article  to  be  plated  is  carefully  cleaned  and 
made  the  cathode  in  a  solution  of  some  salt  of  the  metal  to  be  depos- 


6o6 


ELECTRODYNAMICS 


ited.  The  anode  is  a  plate  of  the  same  metal,  and,  by  dissolving, 
maintains  the  strength  of  the  solution  (Fig.  467).  In  silver  plating 

the  solution  is  a 
complex  cyanide  of 
potassium  and  sil- 
ver; in  gold  plating, 
a  complex  cyanide 
of  potassium  and 
gold;  and  in  nickel 
plating,  a  double 
sulphate  of  nickel 
and  ammonia. 
The  last  is  com- 
FIG.  467.  — Electroplating.  monly  known  as 

nickel  salts,  and 

can  be  bought  in  the  market.  The  details  in  electroplating  differ 
with  the  different  metals,  and  they  must  always  be  attended  to 
with  great  care,  in  order  to  secure  a  smooth  and  coherent  deposit. 
The  pupil  who  wishes  to  try  electroplating  on  a  small  scale 
should  first  acquaint  himself  with  the  details  by  reading  up  on  the 
subject. 

The  use  of  the  electric  current  in  the  treatment  of  ores  and  min- 
erals and  in  refining  metals  is  termed  electrometallurgy.  The  larg- 
est industry  of  this  character  is  the  refining  of  copper.  The  process 
is  similar  to  that  descibed  under  electrotyping.  "  The  crude  copper 
produced  by  the  ordinary  smelting  processes  is  cast  into  heavy  plates 
which  are  used  as  anodes  in  depositing  vats.  The  solution  in  these 
vats  is  copper  sulphate  with  a  little  sulphuric  acid.  The  cathodes 
at  first  are  thin  sheets  of  pure  copper,  but  they  grow  by  deposition 
into  thick  plates  of  copper,  .which  may  be  worked  into  bars  or  drawn 
into  wires  as  desired."  The  current  is  supplied  by  large  dynamos. 
"Other  metals,  such  as  gold,  silver,  and  lead,  are  extracted  from 
their  ores  and  purified  by  electricity,  though  the  older  processes  are 
still  used.  All  the  aluminum,  magnesium,  and  sodium  of  commerce 
are  now  manufactured  by  passing  an  electric  current  through  their 
fused  compounds." 

490.  The  Secondary  or  Storage  Cell.  —  When  a  current  is  sent 
through  an  electrolytic  cell  containing  dilute  sulphuric  acid  and  lead 


CHEMICAL  EFFECTS  OF  CURRENTS  607 

electrodes,  the  electrolysis  of  the  liquid  liberates  oxygen  at  the  anode 
and  hydrogen  at  the  cathode.  The  hydrogen  gathers  in  small  bubbles 
and  escapes.  The  oxygen,  or  a  part  of  it,  combines  with  the  anode, 
forming  a  brown  layer  of  lead  peroxide  (PbO2)  upon  its  surface. 
When  the  anode  is  in  this  condition  the  cell  is  said  to  be  charged,  and 
is  itself  capable  of  generating  an  electric  current.  This  can  be  shown 
by  disconnecting  the  electrodes  from  the  charging  battery  and  con- 
necting them  with  a  galvanometer.  It  will  further  be  found  that 
the  direction  of  the  current  generated  by  the  cell  is  opposite  to  that 
of  the  charging  current;  hence  the  positive  plate  of  the  cell  is  the  one 
that  receives  the  deposit  of  peroxide. 

This  experiment  illustrates  the  principle  of  the  secondary  or 
storage  cell.  If  the  electrodes  are  several  inches  square,  the  current 
from  the  cell  will  probably  be  sufficient  to  ring  an  electric  bell  or  run 
a  small  motor;  but  only  for  a  moment.  While  the  cell  is  generating 
a  current,  hydrogen  ions  of  the  acid  go  to  the  positive  plate,  and  unite 
with  oxygen  from  the  peroxide,  reducing  it  to  monoxide  (PbO).  The 
monoxide  reacts  with  the  acid,  forming  lead  sulphate  (PbSO4),  and 
the  latter  remains  as  an  insoluble  deposit  on  the  surface.  At  the 
same  time  the  sulphions  go  to  the  negative  plate,  with  which  they 
combine,  forming  a  layer  of  lead  sulphate  upon  it.  When  the  plates 
have  thus  been  brought  to  the  same  condition,  the  cell  is  exhausted. 
It  can  be  charged  again  by  means  of  a  current,  as  before.  It  should 
be  noted  that  the  charging  current  does  work  of  chemical  decom- 
position within  the  cell  and  stores  chemical  potential  energy,  not 
electricity.  The  essential  difference  between  the  storage  cell  and 
ordinary  or  primary  cells  is  that  in  the  former  the  chemical  actions 
are  reversible,  and  hence  the  materials  of  the  cell  can  be  used  over 
and  over  indefinitely. 

By  repeatedly  charging  and  discharging  the  cell  used  in  the  above 
experiment,  the  chemical  action  extends  to  an  increasing  depth 
below  the  surface  of  the  lead  plates,  and  the  cell  becomes  capable  of 
receiving  a  greater  charge.  When  the  cell  is  charged,  the  layer  of 
active  material  on  the  negative  plate  is  pure  lead  in  a  spongy  condi- 
tion, and  that  on  the  positive  plate  is  a  porous  crust  of  lead  peroxide. 
In  the  manufacture  of  storage  cells  the  active  material  is  generally 
formed  from  a  paste  made  of  one  or  more  oxides  of  lead  mixed  with 
dilute  sulphuric  acid.  This  paste  is  firmly  imbedded  in  the  openings 
of  a  lead  grid  which  forms  the  body  of  the  plate  (Fig.  468).  Cells  of 


6o8 


ELECTRODYNAMICS 


FIG.  468.  —  Storage  Cell. 


large  capacity  have  several  positive  plates  joined  together,  alternat- 
ing with  negative  plates,  also  joined  together,  as  shown  in  the  figure. 
The  solution  is  one  fourth  (by  weight) 
of  pure  sulphuric  acid  and  three  fourths 
distilled  water.  The  E.M.F.  of  the  cell, 
when  fully  charged,  is  2.2  volts.  It  slowly 
falls  during  the  discharge  to  1.8  volts,  and 
from  that  point  on  the  drop  is  rapid,  with 
more  or  less  permanent  damage  to  the 
cell.  Hence  the  regular  practice  is  to 
discharge  only  to  1.8  volts.  Storage  bat- 
teries are  charged  from  direct-current 
dynamos,  or,  if  from  alternating  current 
circuits,  the  current  is  first  rectified,  i.e., 
changed  to  direct,  by  sending  it  through 
a  special  device  for  that  purpose.  A  good  storage  battery  gives  out 
in  useful  service  about  80%  of  the  energy  expended  in  charging  it. 

The  storage  battery  is  principally  used  as  an  auxiliary  source  of 
power  in  electric  lighting  and  power  stations.  The  battery  is  brought 
into  service  to  supplement  the  output  of  the  dynamos  during  those 
hours  of  the  day  or  night  when  the  demand  for  power  is  greatest,  as 
during  the  early  night  hours  on  lighting  systems;  and  is  charged  dur- 
ing those  hours  when  the  engines  and  dynamos  would  otherwise  be 
idle  or  working  on  light  load.  A  single  cell  of  a  battery  for  such  pur- 
poses, complete  with  plates  and  acid,  weighs  from  200  to  7000  lb., 
according  to  the  size  and  number  of  the  plates.  Storage  batteries 
are  also  used  for  running  electric  launches  and  automobiles.  Their 
great  weight  is  a  hindrance  to  their  more  general  adoption  for  such 
uses,  and  inventors  have  long  sought  to  perfect  a  type  of  cell  that 
would  store  a  much  greater  amount  of  energy  for  a  given  weight. 
The  new  nickel-iron  storage  battery  of  Thomas  A.  Edison  is  a  great 
advance  in  this  direction,  as  it  reduces  the  weight  one  half. 


CHAPTER  XIV 
RADIATIONS.    THE  ELECTRICAL  NATURE  OF  MATTER 

491.  Introduction.  —  We  have  now  compassed  the  field 
of  elementary  physics.  Much  has  been  omitted  that 
might  well  receive  attention,  if  time  were  available  for  it; 
but,  at  least,  no  department  of  the  subject  has  been  slighted. 
The  present  chapter  is  in  the  nature  of  a  sequel,  and  is 
also,  in  a  sense,  an  introduction,  inviting  the  student  to 
more  advanced  fields  of  study. 

Viewing  the  subject  in  retrospect,  it  is  evident  that 
physics  depends  from  first  to  last  upon  the  physical  prop- 
erties of  matter,  using  the  term  matter  in  the  broadest 
sense  to  include  the  ether.  Thus  we  have  the  mechanics 
of  solids,  liquids,  and  gases,  the  mechanics  of  sound  and 
sounding  bodies,  and  molecular  physics  including  heat, 
all  of  which  depend  upon  the  physical  properties  of  ordi- 
nary matter;  while  the  remaining  branches,  light,  magnet- 
ism, and  electricity,  involve  primarily  the  properties  of 
the  ether.  Between  ordinary  matter  and  the  ether  we 
find  an  impassable  barrier  —  neither  can  be  converted  into 
the  other;  yet,  under  certain  conditions,  they  exert  a  mutual 
action  by  which  the  energy  of  either  can  be  transferred 
to  the  other,  as  in  the  emission  and  absorption  of  radiant 
energy,  the  transformations  of  electrical  energy,  etc.  The 
primary  facts  of  the  material  universe  are  summed  up  in 
the  terms  matter,  ether,  electricity,  and  energy.  These  are 
the  actors  in  the  drama  of  nature. 

609 


6io  RADIATIONS 

Elementary  physics  is  mainly  concerned  with  the  scenes 
and  incidents  of  this  drama  (phenomena  and  processes), 
rather  than  with  the  personality  (real  nature)  of  the  actors 
themselves.  But  the  scientist  is  not  content  to  stop  here. 
He  seeks  to  know  what  matter  and  ether  and  electricity 
are,  and  what  the  invisible  mechanism  is  by  which  they  act 
upon  one  another.  If  matter  is  composed  of  atoms,  what 
is  the  atom  and  how  do  atoms  differ?  Are  there  in  fact 
two  kinds  of  electricity,  positive  and  negative,  or  does 
positive  denote  an  excess  and  negative  a  deficiency  of  one 
and  the  same  thing?  Whichever  may  be  true,  the  question 
still  remains,  What  is  electricity,  and  what  is  its  relation 
to  ordinary  matter  and  to  the  ether? 

Science  has  not  arrived  at  the  full  and  final  answer  to 
any  of  these  questions.  Nevertheless  a  wonderful  advance 
in  this  direction  has  been  made  in  recent  years,  and  it  is 
with  this  advance  that  the  present  chapter  mainly  deals. 

I.  SPECTRA  AND  SPECTRUM  ANALYSIS 

492.  Continuous  and  Bright-line  Spectra.  —  One  very 
important  source  of  information  about  matter  is  the 
character  of  the  light  which  it  emits  when  heated  to 
incandescence  in  the  gaseous  state.  The  light  from  all 
incandescent  solids  and  liquids  gives  a  continuous  spec- 
trum (Art.  354),  showing  the  presence  of  all  wave  lengths 
between  the  longest  and  the  shortest.  Hence  there  is 
nothing  in  the  spectra  of  incandescent  solids  and  liquids 
which  serves  to  distinguish  one  substance  from  another. 
With  luminous  vapors  and  gases,  however,  the  case  is 
very  different.  Their  spectra  consist  of  bright,  colored  lines 
(isolated  images  of  the  slit),  separated  by  black  spaces,  and 
no  two  of  these  spectra  are  alike.  Thus  the  spectrum  of 
sodium  vapor  is  a  single  yellow  line  (or  a  very  close 


SPECTRA  AND  SPECTRUM  ANALYSIS  611 

double,  with  wide  dispersion);  that  of  lithium  consists 
of  two  lines  in  the  red.  Hydrogen  gives  a  red,  a  blue, 
and  a  violet  line;  phosphorus,  three  lines  in  the  green; 
strontium,  several  lines  in  the  red;  calcium,  many  lines 
extending  through  the  red,  orange,  yellow,  and  green;  and 
so  on.  Every  element,  when  in  the  gaseous  state,  gives  a 
characteristic  bright-line  spectrum  by  which  the  element 
can  be  identified. 

The  great  importance  of  this  fact  in  physics,  chemistry, 
and  astronomy  has  led  to  the  invention  of  various  forms  of 
spectroscopes  and  spectrometers,  by  means  of  which  the 
radiation  from  any  body  can  be  dispersed  into  a  very  pure 
spectrum  and  the  positions  of  its  different  lines  accurately 
determined.  A  spectrometer  may  be  designed  either  for 
direct  observation  and  measurement,  or  for  taking  photo- 
graphs of  the  spectra.  The  plan  of  a  simple  prism  spectro- 
scope is  shown  in  Fig.  469.  The  essential  parts  are  the 
collimator,  the  prism,  and  the  telescope.  The  collimator 


FIG.  469.  — Section  Diagram  of  Spectroscope. 


consists  of  a  metal  tube  having  a  narrow,  vertical  slit,  S, 
at  one  end,  and  an. achromatic  lens  at  the  other.  The  rays 
from  any  point  of  the  slit  emerge  from  the  lens  parallel, 
the  effect  being  the  same  as  if  the  slit  were  a  distant  lumi- 
nous object.  The  light  to  be  analyzed  passes  through  the 
collimator,  is  dispersed  by  the  prism,  and  is  brought  to  a 
focus  by  the  telescope.  The  observer  views  the  spectrum 
through  the  telescope  as  he  would  view  a  distant  object. 


612  RADIATIONS 

It  is  best  to  use  a  spectroscope  in  a  dark  room.  If  the  room  is 
not  darkened,  a  black  screen  should  be  placed  a  short  distance  beyond 
the  collimator  to  shut  out  diffused  sunlight,  unless  it  is  the  spectrum 
of  sunlight  that  is  under  observation.  When  a  common  gas  burner 
is  placed  before  the  slit,  the  spectrum  is  continuous,  for  the  light  of 
the  flame  comes  from  incandescent  particles  of  solid  carbon.  (This 
solid  carbon  is  deposited  in  the  form  of  soot  when  the  flame  plays 
upon  the  surface  of  cold  porcelain  or  metal.)  The  non-luminous 
Bunsen  flame  gives  no  spectrum,  or,  at  the  most,  only  a  very  faint 
one.  A  strip  of  tin  or  a  wire  heated  to  incandescence  in  the  Bunsen 
flame  gives  a  continuous  spectrum.  Spectra  of  metals  which  are 
easily  volatilized  can  be  studied  by  holding  in  the  Bunsen  flame  a 
bit  of  asbestos  which  has  been  dipped  in  a  solution  of  a  salt  of  the 
metal.  A  convenient  holder  is  made  by  fusing  an  end  of  a  short 
piece  of  platinum  wire  in  the  end  of  a  piece  of  glass  tubing.  The 
free  end  of  the  wire  is  wrapped  round  the  asbestos  (Fig.  470).  In 
this  way  we  may  observe  the  bright-line  spectra  of  sodium,  potassium, 

calcium,  strontium,  barium, 

±^==  ==_         \&  a    etc.     (A  holder   should  be 

provided  for  each  solution, 
otherwise  the  asbestos  will 

contain  traces  of  the  different  metals  and  their  spectra  will  appear 
together.) 

The  line  spectra  of  the  metals  which  vaporize  only  at  very  high 
temperatures  are  obtained  from  the  electric  arc  formed  between  rods 
of  the  metal  whose  spectrum  is  required;  or,  instead  of  the  arc,  the 
sparks  from  an  induction  coil  will  serve.  The  spectrum  of  a  gas, 
e.g.  oxygen,  hydrogen,  nitrogen,  etc.,  is  obtained  by  passing  the 
discharge  from  an  induction  coil  through  the  highly  rarefied  gas  in 
a  vacuum  tube  (Art.  500). 

493.   What    the    Bright-light    Spectrum  Teaches. —  It 

will  be  recalled  that  the  different  colors  of  the  spectrum 
are  the  optical  effects  of  different  wave  lengths  of  the  light 
(Art.  355),  and  that  different  wave  lengths  are  due  to  dif- 
ferent rates  of  vibration,  according  to  the  formula  v  =  In 
(Art.  276).  It  follows  that  a  bright  line  in  a  line  spectrum 
is  formed  by  ether  waves  of  one  definite  length,  and  that 


SPECTRA  AND   SPECTRUM  ANALYSIS  613 

the  source  of  these  waves  is  a  body  having  a  fixed  rate  of 
vibration.  It  follows  further  that  the  atom  of  any  ele- 
ment in  the  gaseous  state  is  the  source  of  as  many  differ- 
ent fixed  rates  of  vibration  as  the  number  of  bright  lines 
in  the  spectrum;  and  this  number  varies  from  a  dozen  or 
less  for  several  of  the  elements  to  many  thousand  for  iron 
and  uranium. 

Let  us  mark  well  the  meaning  of  the  last  statement. 
Every  atom  has  a  definite  number  of  natural  rates  of  vibra- 
tion, just  as  a  piano  or  an  organ  has.  The  comparison 
is  not  far-fetched.  On  the  contrary,  it  fails,  if  anything, 
to  do  justice  to  the  atom;  for  the  range  of  the  piano  is  only 
eighty-eight  notes,  or  different  wave  lengths,  while  that  of 
the  iron  or  uranium  atom  is  several  thousand.  Minute  as 
an  atom  is,  the  idea  that  it  is  a  simple,  structureless  body 
must  evidently  be  set  aside  as  wholly  untenable.  Judged 
by  their  spectra,  atoms  must  be  very  complex  bodies  indeed, 
differing  widely  among  themselves  in  this  respect,  accord- 
ing to  their  kind;  but  on  the  average  they  are  seemingly 
quite  as  complex  as  musical  instruments.  Other  facts 
which  are  presently  to  be  considered  will  offer  some  sug- 
gestions as  to  what  this  complex  structure  may  be. 

The  continuous  spectrum  of  a  substance  in  the  solid  or 
the  liquid  state  is  presumably  due  to  forced  vibrations  of 
the  molecules,  resulting  from  their  frequent  collisions  with 
one  another.  These  forced  vibrations,  owing  to  their  irreg- 
ular character,  give  rise  to  light  waves  of  all  lengths.  In 
gases  and  vapors,  where  collisions  are  relatively  infrequent, 
there  is  a  preponderance  of  free  vibrations  in  the  various 
natural  rates  of  the  molecules  or  their  constituent  atoms. 

494.  Absorption  Spectra.  —  There  is  still  a  third  class 
of  spectra,  due  to  selective  absorption  by  the  medium 


614  RADIATIONS 

through  which  the  light  passes.  We  have  seen  that  col- 
ored glass  and  colored  liquids  produce  such  spectra  (Art. 
360),  and  that,  as  a  rule,  the  absorption  includes  broad 
regions  of  the  spectrum,  sometimes  at  either  end,  sometimes 
in  the  central  portion,  and  sometimes  in  two  or  more  places, 
with  bright  areas  between.  A  spectrum  of  this  character 
is  called  the  absorption  spectrum  of  the  body  which  pro- 
duces the  absorption. 

The  absorption  spectra  of  gases  and  vapors  are  of 
special  interest  and  importance,  for  to  this  class  belong 
the  spectra  of  the  sun  and  stars. 

The  general  conditions  necessary  for  producing  the  absorption  spec- 
trum of  a  vapor  in  class-room  or  laboratory  are,  first,  a  source  of  white 
light  at  a  very  high  temperature;  and,  second,  a  flame  at  a  lower  tem- 
perature, in  which  the  substance  is  vaporized,  so  placed  that  the 
light  from  the  source  passes  through  it  before  reaching  the  prism. 
The  electric  arc  is  best  as  the  source,  since  it  is  the  hottest  obtain- 
able; but  the  oxyhydrogen  lime-light  will  serve.  With  the  arc  light 
the  vapor  may  be  produced  in  a  Bunsen  flame;  but  with  the  lime- 
light a  flame  of  lower  temperature  will  be  necessary,  such  as  that  of 
an  alcohol  lamp.  The  spectrum  can  be  projected  upon  a  screen 
with  a  lantern,  or  viewed  through  a  spectroscope.  In  the  latter  case 
the  alcohol  lamp  or  Bunsen  burner  is  placed  near  the  slit,  and  the 
lime  or  arc  light  a  short  distance  behind  it.  The  chloride  or  nitrate 
of  the  metal  whose  spectrum  is  desired  is  vaporized  in  the  flame, 
e.g.  a  little  table  salt  rubbed  into  the  wick  will  give  the  vapor  of 
sodium.  With  the  Bunsen  burner  a  piece  of  asbestos,  dipped  into 
a  solution  of  the  salt,  is  held  in  the  flame  or  wrapped  round  the  top 
of  the  burner.  If  sunlight  is  used  as  the  source,  the  vapor  will,  in 
general,  merely  intensify  some  of  the  lines  already  present.* 

The  absorption  spectrum  of  a  gas  or  vapor  is  always  a 
dark-line  spectrum,  differing  from  the  complete  spectrum 
of  white  light  only  in  the  fact  that  it  is  crossed  by  one  or 

*  The  teacher  will  find  detailed  directions  for  these  and  other  projection  ex- 
periments in  light  in  the  admirable  little  book  on  Light,  by  Lewis  Wright, 
published  by  the  Macmillan  Company. 


SPECTRA  AND   SPECTRUM  ANALYSIS  615 

more  narrow  dark  lines.  Moreover,  these  dark  lines  are 
the  same  in  number  and  occupy  precisely  the  same  positions 
as  the  lines  in  tJte  bright-line  spectrum  of  the  substance.  This 
is  admirably  shown  when  the  emission  and  absorption 
spectra  of  a  gas  are  produced  side  by  side.  The  bright 
lines  of  the  one  join  accurately  with  the  dark  lines  of  the 
other  (Fig.  471).  The  meaning  of  this  is  that  a  gas  or 
vapor  absorbs  light  of  the  same  wave  lengths  that  it  emits  when 
heated  to  incandescence.  Gaseous  absorption  is  evidently 


FIG.  471. —  Comparison  of  Solar  Spectrum  with  that  of  Iron. 

a  case  of  sympathetic  vibration.  The  atoms  respond  to 
(absorb)  the  vibrations  which  agree  with  their  own  natural 
periods,  just  as  a  tuning  fork  responds  to  the  vibrations  of 
another  fork  of  the  same  pitch  (Art.  287). 

It  follows  from  the  above  that  a  substance  in  the  gaseous 
state  can  be  identified  by  means  of  its  absorption  spectrum 
just  as  certainly  as  by  its  emission  spectrum.  It  is  to  this 
fact  that  the  spectroscope  owes  its  great  importance  in 
astronomical  research. 

495.  The  Solar  Spectrum.  —  It  will  be  remembered  that,  when  a 
fairly  pure  solar  spectrum  is  thrown  upon  a  screen,  it  is  crossed  by  sev- 
eral dark  lines  (Art.  354).  Viewed  through  a  good  spectroscope,  the 
lines  are  more  sharply  defined  and  there  are  many  more  of  them. 
The  greater  the  excellence  and  dispersive  power  of  the  spectroscope 
the  more  numerous  are  the  lines.  Several  thousand  are  shown  in 
an  enlarged  photograph  of  the  solar  spectrum,  over  42  ft.  in  length, 
taken  by  the  late  Professor  Rowland  in  1888.  The  most  prominent 
of  the  lines  are  designated  by  the  letters  from  A  to  H  (Fig.  472). 


6i6 


RADIATIONS 


760| — A— ! 

-a 

6871 — J2 — I 
656 


527 


486 


431 


— D— 


—E— 


T»___ 


Bed  (center  690) 


>  Green(    <« 


530) 


>"Blue( 


The  lines  of  the  solar  spectrum  were  first  carefully  studied  by 
Fraunhofer,  a  noted  German  optician,  in  1814,  and  they  have  since 
been  called  the  Fraunhofer  lines.     Their 
meaning,  however,  remained  a  mystery  until 
the  epoch-making  work  of  the  German  phys- 
\OrangeC  «  600)    icist,  Kirchhoff,  who  developed  the  science 
}Yellow(  «  580)    of  spectrum  analysis,  and  applied  it  to  the 
heavenly  bodies,  during  the  years   1858- 
1862. 

According  to  the  principles  enunciated 
by  Kirchhoff  and  briefly  outlined  in  the 
preceding  pages,  the  Fraunhofer  lines  are 
tt  470)  due  to  absorption  in  some  gaseous  medium 
between  the  sun's  surface  and  the  earth. 
Now  the  only  gaseous  media  which  inter- 
vene are  the  atmospheres  of  the  sun  and 
the  earth;  and,  after  making  due  allowance 
f-Violet(  »  4ltft  for  absorption  in  the  earth's  atmosphere, 
the  balance  must  be  attributed  to  the  sun 
itself.  Considering  the  intense  heat  of  the 

Fl?'-  472TJ;  rQU,nh£fer    sun>  it  is  certain  that  the  elements  known 
Lines  of  the  Solar  Spec- 
trum.   The  numbers  are    upon  the  earth,  if  they  exist  in  the  sun  at 

all,  must  be  present  as  vapors  in  the  solar 
atmosphere.  The  Fraunhofer  lines  are  un- 
impeachable witnesses  to  the  truth  of  this 
conclusion.  Over  two  thousand  of  these  lines  are  found  to  coincide 
in  position  with  the  bright  lines  in  the  gas  spectrum  of  iron  (Fig. 
471),  proving  that  iron  exists  as  a  vapor  in  the  sun's  atmosphere. 
The  D  line  of  the  spectrum  coincides  with  the  yellow  line  of 
sodium.  The  lines  C  and  F  are  due  to  hydrogen.  Similar  identifi- 
cations have  been  made  for  about  half  of  the  known  terrestrial 
elements,  including  iron,  nickel,  cobalt,  carbon,  calcium,  magnesium, 
sodium,  and  hydrogen. 

Wonderful  indeed  is  the  fact  that  an  element  in  the  far-distant 
sun  is  thus  able  to  reveal  its  existence  to  us,  just  as  certainly  as  if  we 
had  a  sample  for  examination  in  the  chemical  laboratory.  The  story 
of  the  stars  is  also  written  in  their  light  and  revealed  in  the  dark  lines 
of  their  spectra  after  the  same  fashion,  although  the  ether  waves 
which  carry  the  message  may  have  been  traveling  for  hundreds  of 


•"fcfc 


the  wave  lengths,  ex- 
pressed in  millionths  of 
a  millimeter. 


ELECTRIC  OSCILLATIONS  AND  WAVES          617 

years.  Nor  is  this  all.  The  spectrum  of  a  star  shows  whether  the  star 
is  moving  toward  or  from  the  earth  and  at  what  rate.  For  motion 
toward  the  earth  has  the  effect  of  shortening  all  light  waves,  since 
more  of  them  reach  the  earth  in  a  given  time  than  would  be  the  case 
if  the  distance  were  constant.  The  result  is  a  slight  shifting  of  all 
the  dark  lines  toward  the  violet  end  of  the  spectrum;  and,  from  the 
amount  of  the  displacement,  the  velocity  of  the  star  can  be  computed. 
Motion  from  the  earth  has  the  contrary  effect,  and  all  the  lines  are 
displaced  toward  the  red  end  of  the  spectrum.  In  this  way  it  has 
been  found  that  the  brilliant  star  Arcturus  is  rushing  toward  us  at 
the  rate  of  nearly  60  mi.  per  second. 

II.  ELECTRIC  OSCILLATIONS  AND  WAVES.     ELEC- 
TRO-MAGNETIC THEORY  OF  LIGHT 

496.  Electric  Oscillations.  —  The  discharge  of  a  Leyden 
jar  produces  what  appears  to  be  a  single  spark;  but,  when 
viewed  in  a  rapidly  revolving  mirror,  it  is  found  to  consist 
of  a  series  of  sparks,  often  a  dozen  or  more.  If  a  concave 
mirror  is  arranged  to  reflect  the  light  and  focus  it  upon  a 
photographic  plate,  the  impressions  due  to  the  individual 
sparks  of  a  series  are  drawn  out  into  a  line,  owing  to  the 


* 


FIG.  473-  —  Photograph  of  Oscillating  Electric  Sparks. 

slight  angle  through  which  the  mirror  turns  in  the  brief 
intervals  between  their  occurrence  (Fig.  473).  Under 
ordinary  conditions  these  intervals  are  less  than  the 
millionth  part  of  a  second,  as  shown  by  computation 
based  on  the  known  rate  of  rotation  of  the  mirror,  the 
distance  between  it  and  the  photographic  plate,  and  the 
distance  between  the  spark  images  on  the  plate. 

The  meaning  of  the  series  of  sparks  is  that  the  discharge 


6i8 


RADIATIONS 


of  a  Leyden  jar  is  oscillatory.  The  current  surges  back  and 
forth  between  the  inner  and  outer  coats  of  the  jar,  gradu- 
ally dying  away,  just  as  a  pendulum  or  a  spring  executes 
a  series  of  vibrations  with  diminishing  amplitude  before 
coming  to  rest.  The  period  of  the  oscillations  is  deter- 
mined by  the  capacity  of  the  jar  and  the  resistance  and 
self-induction  of  the  discharging  circuit.  The  smaller 
these  factors  are  the  shorter  will  be  the  period.  When  two 
small  metal  spheres  are  substituted  for  the  jar,  and  the 
discharge  takes  place  by  a  short,  straight  path  between 
them,  the  period  may  be  less  than  one  hundred-millionth 
of  a  second. 

497.  Electromagnetic  Waves.     Electrical  Resonance.  - 

It  can  be  shown  in  various  ways  that  electric  oscillations 
produce  waves  in  the  ether,  which  radiate  from  the  center 
of  disturbance  with  the  velocity  of  light  (300,000,000 
meters  per  second).  The  existence  of  such  waves  is  demon- 
strated in  the  following  experiment,  due  to  the  English 
physicist,  Sir  Oliver  Lodge. 

Two  Leyden  jars,  A  and  B  (Fig.  474),  of  equal  capacity  are  con- 
nected with  discharge  circuits,  each  consisting  of  a  wire  rectangle. 

A 's  circuit  has  a  spark  gap, 
S,  between  two  metal  balls. 
B's  circuit  is  without  gap 
between  the  coats  of  the  jar, 
and  its  size  is  adjustable  by 
means  of  the  sliding  wire  M . 
A  strip  of  tin-foil  extends 
from  the  inner  coat  of  B  to 
FIG.  474.  — Apparatus  for  Showing  Electri-  with;n  I  mm.  of  the  outer 
cal  Resonance.  m,  .  ., 

coat  at  e.    The  two  circuits 

are  placed  parallel  to  each  other,  and  the  coats  of  A  are  connected 
with  the  terminals  of  an  induction  coil  or  with  the  opposite  sides 
of  an  electrostatic  machine.  When  A  discharges,  a  spark  occurs 


ELECTRIC  OSCILLATIONS  AND  WAVES          619 

at  e,  provided  the  areas  included  within  the  two  rectangles  are  equal 
or  nearly  so;  but  there  is  no  response  at  e  if  the  area  of  B's  circuit 
is  made  considerably  larger  or  smaller  than  A's. 

This  is  explained  as  follows.  Each  circuit  has  a  natural 
period  of  oscillation,  which  is  determined  by  the  capacity 
of  the  jar  and  the  self-induction  of  the  circuit.  (The 
resistance  of  the  circuit  is  negligible.)  The  capacities  are 
equal,  and  the  self-induction  increases  with  the  area  of 
the  rectangle.  Hence  with  equal  areas  the  periods  are 
equal,  and  B 's  circuit  responds  to  an  oscillatory  discharge 
in  A ,  just  as  a  tuning  fork  responds  to  another  of  the  same 
pitch.  The  phenomenon  is  termed  electrical  resonance, 
and  the  two  circuits  are  said  to  be  in  tune  with  each  other. 
The  induced  oscillations  are  set  up  through  the  medium  of 
the  ether,  which  transmits  the  impulses  in  the  form  of  elec- 
tromagnetic waves.  When  the  circuits  are  not  in  tune, 
the  induced  oscillations  are  relatively  weak,  —  too  weak  to 
produce  a  spark  at  e  if  the  difference  between  the  periods 
is  considerable. 

498.  The  Electromagnetic  Theory  of  Light.  —  As  early 
as  1867  one  of  England's  greatest  physicists,  James  Clerk 
Maxwell,  advanced  the  theory  that  light  is  an  electromag- 
netic rather  than  a  simple  mechanical  disturbance  of  the 
ether,  and  that  it  ought  to  be  possible  to  produce  waves  of 
this  character  by  electrical  means.  Twenty-one  years 
elapsed  before  his  theory  was  experimentally  verified  by 
the  German  physicist,  Heinrich  Hertz.  Hertz  demon- 
strated not  only  that  such  waves  were  produced  by  an 
oscillatory  spark  discharge,  but  also  that  they  possess  all 
the  properties  of  light  waves,  being  reflected,  refracted, 
etc.,  according  to  the  same  laws.  The  only  essential 
difference  is  in  the  length  of  the  waves.  For  the  longest 


620  RADIATIONS 

visible  waves  (extreme  red),  this  is  only  .000076  cm.;  while 
the  ordinary  length  of  electric  waves  is  several  meters, 
and  the  shortest  yet  produced  were  .4  cm. 

Since  the  velocity  of  all  ether  waves  is  the  same,  their 
frequencies  are  inversely  proportional  to  their  wave  lengths 
(by  the  formula  v  =  In).  If  the  frequency  of  an  electrical 
oscillation  is  100,000,000  per  second,  it  will  send  out  electric 

waves  of  the  length  - — - =  3  m.     Conversely  the 

100,000,000 

frequency  of  the  longest  visible  waves  is  —  —  >  or, 

.00000076 

very  nearly  400,000,000,000,000  vibrations  per  second. 
Oscillations  at  such  a  stupendous  rate  must  of  necessity 
be  on  an  inconceivably  small  scale.  As  we  shall  presently 
see,  there  are  substantial  reasons  for  believing  that  the 
individual  source  of  light  waves  is  a  vibrating  particle  of 
electricity,  or  a  group  of  such  particles,  within  the  atom. 

499.  Wireless  Telegraphy.  —  Since  1895  many  systems  of  wire- 
less telegraphy  have  been  devised  for  transmitting  signals  through 
space  by  means  of  electric  waves.  The  sending  mechanism  is  de- 
signed to  produce  electric  oscillations  which  are  under  the  control  of 
the  operator.  For  long-distance  working,  the  apparatus  must  be 
very  powerful,  since  the  waves  spread  out  in  all  directions,  and  grow 
rapidly  weaker  as  they  travel.  The  receiving  apparatus  must  be 
as  sensitive  as  possible,  for  the  same  reason.  So  well  have  these 
requirements  been  met  that  messages  have  been  sent  over  distances 
exceeding  3000  mi. 

The  details  of  construction  and  operation  of  any  fully  developed 
system  of  wireless  telegraphy  are  numerous  and  complicated,  and  the 
principles  involved  are  largely  beyond  the  scope  of  this  book.  It  is, 
however,  a  comparatively  simple  matter  to  demonstrate  the  main 
fact,  viz.,  that  intelligible  signals  can  be  transmitted  by  means  of 
ether  waves.  This  is  shown  in  the  following  experiment. 

The  transmitter  or  oscillator  is  a  small  induction  coil,  with  a  plate 
of  sheet  metal  and  a  discharge  ball  connected  to  each  of  the  secondary 


ELECTRIC  OSCILLATIONS  AND  WAVES 


621 


terminals  (Fig.  475).  The  plates  are  charged  by  the  coil,  and  they 
discharge  with  an  oscillatory  spark  across  the  gap.  The  current 
is  controlled  by  means  of  a 
telegraph  key  in  the  primary 
circuit. 

One  essential  part  of  the 
receiving  apparatus  is  some 
device  that  is  very  sensitive 
to  electric  waves.  The  co- 
herer, C  (Fig.  476),  was  first 
used  for  this  purpose  in  in-  FlG-  475- -Hertz  Oscillator  for  Transmitting 
dustrial  wireless  systems, 

and  will  serve  for  our  experiment.  It  consists  of  a  glass  tube  of 
small  bore,  containing  a  small  quantity  of  metal  filings  (silver,  nickel, 
or  iron)  and  two  wire  electrodes.  The  filings,  lying  loosely  between 

the  electrodes,  ordinarily 
offer  a  very  great  resistance 
to  the  passage  of  a  current; 
but  the  loose  mass  suddenly 
becomes  a  good  conductor 
under  the  action  of  electric 
waves.  Apparently  the 
waves  cause  the  filings  to 
FIG.  476.— Receiving  Apparatus  for  Wireless  cjmg  together,  thus  reducing 
Telegraphy.  ^  resistance  at  their  points 

of  contact.  Jarring  the  tube  restores  the  filings  to  their  original  con- 
dition of  high  resistance. 

A  diagram  of  the  complete  receiving  apparatus  is  shown  in  Fig. 
476.  The  coherer,  C,  connects  two  metal  plates,  PP,  of  the  same 
size  as  those  of  the  transmitter.  The  coherer  is  also  included  in  cir- 
cuit with  a  battery  cell  B  and  the  magnet  coils  of  a  relay  R.  This 
relay  controls  a  second  battery  circuit,  connected  with  an  electric 
bell.  The  bell  is  so  placed  that  the  clapper  strikes  the  coherer  on 
the  back  stroke,  thus  acting  as  a  decoherer. 

When  the  key  of  the  transmitter  is  depressed,  sparks  pass  between 
the  knobs  of  the  oscillator,  and  electric  waves  are  sent  out.  These, 
falling  upon  the  plates  of  the  receiver,  set  up  oscillations  of  the  same 
frequency  between  them.  This  breaks  down  the  resistance  of  the 
coherer  and  permits  the  passage  of  a  battery  current  through  the 


622  RADIATIONS 

relay.  The  relay  closes  the  bell  circuit;  the  bell  rings;  and  the 
clapper,  on  the  return,  strikes  the  coherer.  The  apparatus  is  then  in 
condition  to  receive  another  signal. 

In  commercial  wireless  systems  the  waves  are  sent  out  and  received 
by  means  of  an  aerial  wire  or  wires,  carried  up  to  a  height  of  100  to 
200  ft.  on  a  mast.  For  sending,  this  wire  is  connected  with  one 
terminal  of  the  spark  gap.  The  other  terminal  of  the  spark  gap  is 
connected  with  the  ground.  For  receiving,  the  coherer  or  other 
sensitive  detector  of  electric  waves  takes  the  place  of  the  spark  gap. 
There  are  various  forms  of  detectors  in  connection  with  which  the 
signals  are  received  through  a  high-resistance  telephone  receiver. 

III.  ELECTRIC  CONDUCTION    THROUGH  GASES,  CATHODE 

AND    R6NTGEN   RAYS 

500.  Electric  Discharge  in  Rarefied  Gases.  Geissler 
Tubes.  —  The  study  of  the  electric  discharge  in  rarefied 
gases  has  led  to  results  of  very  great  theoretical  and  practi- 
cal importance ;  and  the  beauty  and  novelty  of  the  phenom- 
ena never  fail  to  arouse  the  liveliest  interest.  To  study 
the  effect  of  different  pressures  we  may  use  a  glass  tube, 
20  cm.  or  more  in  length,  with  a  side  connection  for  exhaust- 
ing the  air,  and  platinum  electrodes,  A  and  C,  sealed  into 
the  ends  (Fig.  477).  The  tube  is  connected  with  the  ter- 
^or*G<r  minals  of  an  induction 

coil  or  a  static  machine, 
with  a  spark  gap  of  two 
or  three  inches  between 
the  knobs.  Before  the 
air  is  exhausted  from 

FIG.  477-- Vacuum  Tube.  ^    ^   ^   discharge 

takes  place  between  the  knobs,  since  the  shorter  path  offers 
the  less  resistance;  but,  when  the  pressure  is  reduced  to  a  few 
centimeters  of  mercury,  the  discharge  takes  place  within 
the  tube.  From  this  we  know  that  the  resistance  of  a 


ELECTRIC  CONDUCTION  THROUGH  GASES      623 

gas  diminishes  rapidly  with  _th.e_pressure.  Moreover,  the 
appearance  of  the  discharge  undergoes  a  striking  change. 
It  is  no  longer  a  brilliant,  irregular,  narrow  line  of  light, 
as  in  the  open  air,  but  a  pale  violet  or  crimson  glow,  extend- 
ing along  the  axis  of  the  tube.  As  the  air  is  further  ex- 
hausted, the  luminous  band  broadens  out  until  it  fills  the 
tube,  becoming  paler  in  tint  and  nebulous  in  form.  In  a 
dark  room  the  effect  is  very  beautiful. 

At  a  pressure  of  i  mm.  of  mercury,  or  slightly  less,  the  general 
appearance  of  the  discharge  is  that  shown  in  Fig.  478.  A  thin  layer 
of  velvety  light  envelops  the  cathode.  Surrounding  this  is  the 
Crookes  dark  space,  then  the  negative  glow,  and  beyond  this  a  second 
dark  space.  Between  the  latter  and  the  anode  there  is  a  luminous 
region,  called  the  positive  column.  The  anode  itself,  if  it  is  a  wire, 
has  only  a  small  bright  star  of  light  at  the  end.  At  a  certain  pressure 
the  positive  column  breaks  up  into  a  set  of  stria,  or  patches  of  light 
of  cup-like  form,  which  vibrate  to  and  fro  between  darker  spaces 
(Fig.  478).  The  color  of  the  striae  depends  on  the  nature  of  the  gas 
within  the  tube.  Vacuum  tubes,  designed  especially  to  give  beauti- 
ful effects,  are 
made  in  a  great  va- 
riety of  shapes,  and 
often  of  certain 
kinds  of  glass  which 
glow  with  a  phos- 
phorescent  light 

when  the  discharge  .        ,,  ~  .   .     ,_  . 

FIG.  478.  —  Geissler  Tube, 
passes.    These    are 

known  as  Geissler  tubes,  after  a  German  glass  blower  who  became 
noted  for  his  skill  in  making  them.  Tubes  for  spectrum  analysis 
are  of  simple  design,  and  contain  traces  of  the  gas  whose  spectrum 
is  desired. 

The  weird  phenomenon,  known  as  the  aurora  borealis  or  northern 
lights  (Fig.  479),  is  evidently  due  to  electrical  discharges  in  the  upper 
air,  similar  to  the  discharge  in  a  Geissler  tube.  In  high  northern 
latitudes  the  aurora  is  frequently  seen  in  winter,  and  within  and  near 
the  Arctic  circle  it  is  of  almost  nightly  occurrence.  Its  usual  form 


624  RADIATIONS 

"  is  that  of  a  number  of  ill-defined  streaks  or  streamers  of  pale  tint 
(sometimes  tinged  with  red  and  other  colors),  either  radiating  in  a 
fanlike  form  from  the  horizon  in  the  direction  of  the  magnetic  north, 
or  forming  a  sort  of  arch  across  that  region  of  the  sky.  A  certain 
flickering  or  streaming  motion  is  often  discernible  in  the  streaks. 
Under  very  favorable  conditions  the  aurora  extends  over  the  entire 
sky."  A  similar  phenomenon,  termed  the  aurora  australis,  is  seen 
in  the  south  polar  regions.  Auroras  are  most  brilliant  during  mag- 
netic storms,  and  there  is  evidently  some  connection  between  them; 
but  the  precise  conditions  which  give  rise  to  either  are  only  a  matter 
of  conjecture. 


FIG.  479.  —  Aurora  Borealis. 

501.  Crookes'  Tubes  and  Cathode  Rays.  —The  study  of 
the  electric  discharge  in  a  moderate  vacuum,  as  in  Geissler 
tubes,  came  to  little  of  real  value  beyond  its  application 
in  spectrum  analysis;  but  a  very  important  field  of  in- 
vestigation was  opened  up  when  Sir  William  Crookes,  of 
England,  began  to  experiment  with  tubes  in  which  the 
pressure  was  reduced  to  about  .001  mm.  or,  roughly,  one 
millionth  of  an  atmosphere  (1873-1878).  By  the  end  of 


ELECTRIC  CONDUCTION  THROUGH  GASES      625 

the  century  this  field  had  been  very  thoroughly  exploited, 
through  the  joint  efforts  of  several  skilled  investigators. 

As  the  pressure  in  a  vacuum  tube  is  decreased  to  .01  mm. 
or  less,  the  glow  within  it  gradually  vanishes,  and  an  invis- 
ible radiation  proceeds  from  the  cathode.  The  cathode 
rays,  as  they  are  termed,  disclose  their  presence  in  various 
ways.  Where  they  strike  the  wall  of  the  tube  it  is  made 
luminous  and  the  temperature  rises.  The  color  of  the  light 
which  is  thus  produced  varies  with  the  kind  of  glass;  with 
ordinary  glass  it  is  greenish  yellow.  The  cathode  is  usu- 
ally a  flat  or  concave  metal  disk  (Figs.  480-482);  and  the 
rays  proceed  from  it  in  straight  lines  perpendicular  to  Us  sur- 
face, regardless  of  the  position  of  the  anode.  This  is  proved 
by  the  sharp  shadow  cast  on  the  walls  of  the  tube,  where  the 
rays  are  intercepted  by  a  screen  (Fig.  480).  (This  behavior 
is  in  strong  contrast  with  that  of  the  luminous  path  of  the 
discharge  in  a  moderate 
vacuum,  which  curves 
to  meet  the  anode  wher- 
ever the  latter  may  be 
placed.) 

The  heating  effect  of 
the  rays  is  splendidly 
shown  in  a  "  focus  tube  " 
(Fig.  482),  in  which, 
owing  to  the  perpendic- 
ular projection  of  the  rays,  they  are  brought  to  a  focus 
upon  a  platinum  disk,  Z>,  by  the  concave  cathode,  C.  The 
disk  becomes  red  hot  in  a  few  minutes. 


FIG.  480. —  Shadow  Cast  by  Mica  Cross  in 
Cathode  Rays. 


602.   Nature  of  the  Cathode  Rays.     Electrons.  —  The 

nature  of  the  cathode  rays  has  been  determined  from  an 
exhaustive  study  of  their  properties.     It  is  found  that: 


626  RADIATIONS 

1.  They  are  deflected  by  a  magnetic  field. 

2.  They  are  deflected  by  an  electrostatic  field. 

3.  They  impart  a  negative  charge  to  an  insulated  con- 
ductor upon  which  they  fall  inside  the  tube. 

4.  They  are  stopped  by  the  glass  walls  of  the  vacuum 
tube,  but  they  can  pass  through  a  very  thin   plate  of 
aluminum. 

The  first  three  of  these  properties  demonstrate  that  the 
cathode  radiation  does  not  consist  of  ether  waves  (as  some 
at  first  supposed),  but  of  particles  projected  from  the  sur- 
face of  the  cathode.  A  moving  electric  charge  is  virtually 
an  electric  current;  and  it  sets  up  a  magnetic  field  along 
its  path,  just  as  an  electric  current  in  a  wire  does.  And 
since  a  wire  in  which  a  current  is  flowing  tends  to  move 
sideways  across  the  lines  of  force  in  a  magnetic  field  (as  in 
the  motor  and  the  D' Arson val  galvanometer),  it  follows 
that  a  charged  particle,  shooting  across  lines  of  force  in 
a  magnetic  field,  must  be  acted  upon  by  a  sideward  force 
which  tends  to  deflect  it  from  a  straight  path.  This  con- 
clusion tallies  exactly  with  the  observed  fact  (Fig.  481). 
Similarly  it  is  found  that  the  cathode  rays  are  deflected 

when  they  pass  through 
an  electrostatic  field 
between  two  plates, 
one  of  which  is  posi- 

FIG.  481.— Deflection    of    Cathode    Rays    in    tively    and    the    Other 

a  Magnetic  Field.  negatively  charged. 

These  experiments,  and  others  which  are  too  elaborate 
to  be  considered  here,  afford  the  necessary  data  from  which 
it  is  possible  to  compute  the  mass  and  velocity  of  the  cath- 
ode-ray particles,  and  the  magnitude  of  the  charge  which 
they  carry.  It  turns  out  that  the  mass  of  a  particle  is  about 
of  the  mass  of  a  hydrogen  atom,  which  is  the  light- 


ELECTRIC  CONDUCTION  THROUGH  GASES      627 

est  atom  of  all  the  elements.  The  hydrogen  atom  thus 
gives  place  to  the  cathode-ray  particle  as  the  smallest 
thing  known  in  the  universe.  The  charge  of  a  particle  is 
equal  to  the  smallest  electrical  charge  carried  by  ions  in 
electrolysis  (Art.  487).  The  velocity  of  the  particles  varies 
somewhat,  but  the  average  is  roughly  one  tenth  of  the  veloc- 
ity of  light,  or  from  15,000  to  20,000  mi.  per  second,  or 
about  60,000  times  the  velocity  of  a  rifle  ball! 

Experiment  further  shows  that  the  mass,  charge,  and 
velocity  of  the  cathode  particle  are  all  independent  of  the 
nature  of  the  gas  in  the  vacuum  tube  or  the  kind  of  metal 
used  for  the  electrode.  Hence  it  is  believed  that  the 
cathode  particle  is  one  and  the  same  thing  in  all  cases. 
It  is  called  an  electron,  or,  sometimes,  a  negative  corpuscle 
or  negative  ion. 

The  most  remarkable  property  of  the  electron  is  yet  to 
be  mentioned.  A  moving  electrical  charge  possesses  inertia 
or  mass,  which  appears  to  be  identical  in  kind  with  the 
mass  of  ordinary  matter;  for  the  moving  charge  has  both 
momentum  and  kinetic  energy.  Prof.  J.  J.  Thomson,  of 
Cambridge,  England,  has  shown  mathematically  that  the 
observed  behavior  of  electrons  can  be  accounted  for  on  the 
assumption  that  the  whole  of  the  mass  of  an  electron  is 
due  to  the  charge.  On  this  view  an  electron  is  the  dis- 
embodied, indivisible,  natural  unit  or  atom  of  negative  elec- 
tricity —  pure  electricity  and  nothing  else. 

503.  Positive  Ions.  Canal  Rays.  —  When  the  cathode  of  a 
Crookes  tube  is  perforated  with  many  small  holes,  it  is  observed  that, 
in  addition  to  the  cathode  rays,  which  are  emitted  from  one  side, 
there  are  rays  proceeding  in  the  opposite  direction,  apparently 
through  the  holes  in  the  cathode.  These  are  called  canal  rays.  They 
can  be  deflected  by  a  magnetic  or  an  electric  field,  and  the  direction 
of  the  deflection  shows  that  they  carry  a  positive  charge.  They  are, 


628  RADIATIONS 

in  fact,  positively  charged  particles  or  ions,  moving  much  more  slowly 
than  the  electrons.  Unlike  the  electron,  the  mass  of  the  positive 
ion  varies  with  the  material  of  the  electrodes  and  with  the  kind  of 
gas  in  the  discharge  tube.  Apparently  the  mass  is  the  same  as  that 
of  the  atoms  of  the  elements  which  happen  to  be  present  in  the  tube. 
The  charge  of  the  positive  ion  is  believed  to  be  equal  to  that  of  the 
electron  (but  opposite  in  sign). 

Taking  all  the  facts  together,  they  present  a  very  strong  argument 
in  support  of  the  view  that  the  positive  ion  of  a  gas  is  simply  an  atom 
which  has  lost  an  electron.  In  an  electrolyte  (Art.  487),  according 
to  this  view  the  positive  ions  are  atoms  which  have  lost  one,  two 
three,  or  four  electrons,  according  as  their  charge  is  one,  two,  three, 
or  four  times  that  of  the  hydrogen  ion;  and  the  negative  ions  are 
atoms  or  groups  of  atoms  which  have,  for  the  time  being,  appropri- 
ated one  or  more  extra  electrons,  stolen  from  the  positive  ions. 


504.  Rbntgen  or  X-rays.  —  Where  cathode  rays  strike 
the  walls  of  a  Crookes  tube  or  any  solid  within  it,  they  excite 
a  form  of  invisible  radiation  which  is  said  to  consist  of 
Rontgen  or  X-rays.  Rontgen  is  the  name  of  the  German 
physicist  who  discovered  the  rays  in  18*95.  They  were  called 
by  him  X-rays,  because  their  nature  was  unknown. 

Rontgen  rays  proceed  in  straight  lines  from  their  source. 
They  are  not  reflected  or  refracted;  hence  can  not  be  of 
the  nature  of  light  waves.  They  do  not  carry  an  electric 
charge,  and  are  not  deflected  by  a  magnetic  or  an  electric 
field.  They  pass  through  glass,  and  also  through  substances 
which  are  opaque  to  light,  such  as  wood,  thin  sheets  of 
metal,  and  animal  tissues.  They  affect  a  photographic 
plate,  and  excite  fluorescence  (light  from  a  cold  body) 
when  they  fall  on  certain  substances.  It  is  believed  that 
they  are  single,  disconnected  pulses  in  the  ether,  traveling 
with  the  velocity  of  light,  and  that  each  pulse  is  due  to 
the  sudden  stopping  of  a  cathode  particle  when  it  strikes 
a  solid. 


ELECTRIC    CONDUCTION    THROUGH    GASES     629 

In  the  treatment  of  diseased  tissues  of  the  body  by  means  of  X-rays 
and  in  X-ray  photography, 

it  is  an  advantage  to  have  r  ^s-^        ^W4%. 

the  source  of  the  rays  as 
small  as  possible.  The  focus 
tube  (Fig.  482)  is  designed 

with  this  end  in  view.     The  ./#SSSS^^? 

cathode,  C,  is  concave,  in 
order  to  concentrate  its  rays 
upon  the  center  of  the  plati- 
num disk,  D.  The  X-rays  are  generated  at  this  point. 

X-ray  photographs  are  shadow  pictures  (Fig.  483).  The  photo- 
graphic plate  is  kept  in  a  plate  holder  or  wrapped  in  paper  to  protect 
it  from  light,  and  the  object  to  be  photographed  is  placed  against  it. 
The  parts  or  structure  of  the  object  will  be  recorded  in  the  picture 
just  in  so  far  as  the  different  parts  are  unequally  transparent  to  the 
rays.  Since  flesh  is  quite  transparent  and  bones  are  rather  opaque, 


FIG.  482.  — X-ray  Tube. 


FIG   483.  —  X-ray  Photograph  of  a  Broken  Arm. 

a  photograph  of  any  part  of  the  body  shows  the  bony  structure  very 
clearly.  X-ray  photography  is  thus  an  invaluable  aid  to  the  surgeon 
in  determining  the  character  of  the  injury  in  cases  of  fractured  or 
broken  bones,  as  well  as  in  locating  foreign  bodies,  such  as  bullets, 
needles,  etc.,  for  the  metals  are  also  less  transparent  than  the  flesh. 


630  RADIATIONS 

The  same  information  can  be  obtained  directly  by  sight,  with  the 
aid  of  a  fluoroscope  (Fig.  485).     This  is  a  darkened  box,  shaped  at 

one  end  so  as  to  fit  closely  round  the 
eyes  of  the  observer,  and  closed  at  the 
other  end  with  a  fluorescent  screen. 
This  screen  is  covered  with  some 
substance  (usually  barium  platino- 
cyanide)  which  becomes  luminous 
under  the  influence  of  X-rays.  If 

the  hand  is  placed  against  the  screen 
Fig.  484.  —  Fluoroscope.  ,  .,     , ,      ,   ,         .  ,  ,     , , 

while  the  latter  is  exposed  to  the  rays, 

the  shadow  of  the  hand  will  appear  upon  the  screen,  the  flesh  show- 
ing rather  light  and  the  bones  dark. 


IV.  RADIOACTIVITY.    ELECTRICAL  THEORY  OF  MATTER 

505.  Discovery   of   Radioactivity.  —  We  now  come  to 
another    epoch-making    discovery,    which    opened    up    a 
hitherto  unknown  and  unsuspected  field  of  investigation 
of  surpassing  interest.     In  1896  the  distinguished  French 
physicist,  M.  Henri   Becquerel,  found   that   the  element 
uranium  and  all  its  compounds  emit  rays  which  are  able  to 
pass  through  black  paper  and  affect  a  photographic  plate. 
These  rays  are  given  out  continually  and  without  the  aid 
of  any  outside  agency.     Uranium  and  certain  other  ele- 
ments, which  were  later  found  to  possess  the  same  property, 
are  said  to  be  radioactive,  and  the  property  itself  is  termed 
radioactivity. 

506.  Use  of  the  Electroscope  in  the  Study  of  Radioactivity.  —  We 

must  turn  aside  for  a  moment  to  learn  something  of  the  use  of  the 
electroscope  in  these  investigations.  In  a  dry  atmosphere,  under 
normal  conditions,  a  well  insulated  electroscope. (Fig.  367)  retains  its 
charge  for  several  hours.  Now  it  is  found  that  dry  air  can  be  ren- 
dered conducting  in  several  ways;  and,  when  this  happens  within 
an  electroscope,  the  leaves  are  discharged  more  or  less  rapidly,  accord- 
ing to  the  degree  of  conductivity  imparted. 


ELECTRICAL  THEORY  OF  MATTER     631 

This  action  takes  place  when  a  beam  of  rays  from  an  X-ray  tube 
falls  upon  a  charged  electroscope,  as  is  shown  by  the  fact  that  the 
leaves  immediately  begin  to  fall  together.  The  action  of  the  rays  is 
indirect;  for  when  they  pass  through  air  in  a  separate  vessel  and  this 
air  is  afterward  introduced  into  the  electroscope,  the  leaves  are  dis- 
charged. The  theory  is  that  the  X-rays,  in  passing  through  a  gas, 
create  such  an  atomic  disturbance  that  here  and  there  an  atom  loses 
an  electron  (negative  ion),  and  itself  becomes  a  positive  ion.  The 
process  is  called  ionization.  The  gold  leaves  are  discharged  in  ion- 
ized air  by  attracting  to  themselves  the  ions  of  opposite  sign,  —  the 
negative  ions  if  the  charge  on  the  leaves  is  positive,  and  vice  versa. 

The  rays  emitted  by  all  the  radioactive  substances  ionize  the  air 
through  which  they  pass,  and  so  have  the  power  to  discharge  an 
electroscope.  As  a  means  of  detecting  the  presence  of  minute  quan- 
tities of  a  radioactive  element,  the  electroscope  as  far  surpasses  the 
spectroscope  as  the  latter  does  the  most  sensitive  balance,  the  ratio 
of  sensitiveness  in  either  case  being  in  the  neighborhood  of  100,000 
to  one.  "  The  quantity  of  any  radioactive  substance  which  can  be 
detected  is  to  the  corresponding  amount  of  the  other  elements,  which 
can  be  detected  only  by  the  ordinary  methods  of  chemical  analysis, 
as  a  second  is  to  a  thousand  years."  —  /.  /.  Thomson. 

507.  Other  Radioactive  Elements.  —  The  discovery  of 
Professor  Becquerel  brought  other  workers  into  the  field, 
and  an  extended  search  was  made  for  other  radioactive 
substances.  In  1898  thorium  and  its  compounds  were 
added  to  the  number.  Thorium  is  a  constituent  of  the 
Welsbach  gas  mantle.  A  piece  of  such  a  mantle,  pressed 
out  flat  on  a  photographic  plate  and  left  in  the  dark  for  a 
week  or  more,  takes  its  own  photograph. 

Among  the  substances  examined  was  the  mineral  pitch- 
blende, from  which  uranium  is  principally  obtained. 
Madame  Curie,  of  Paris,  found  that  the  activity  of  this  ore, 
as  shown  by  the  rate  of  discharge  of  an  electroscope,  was 
three  or  four  times  as  great  as  that  of  pure  uranium.  It 
was  evident  that  pitchblende  must  contain  some  other  sub- 
stance of  much  greater  radiating  power  than  uranium. 


632  RADIATIONS 

To  extract  this  substance  from  the  ore  proved  to  be  a  most 
difficult  task,  partly  because  it  was  present  only  in  exces- 
sively minute  quantities;  but,  from  a  ton  of  the  ore,  Mme. 
Curie  finally  obtained  a  few  milligrams  of  the  new  element, 
which  she  called  radium  (1898).  The  ray-emitting  power 
of  this  wonderful  element  is  about  1,800,000  times  as  great 
as  that  of  an  equal  quantity  of  uranium. 

From  the  same  ore  Mme.  Curie  extracted  another  highly  radio- 
active substance,  which  she  named  polonium;  and  M.  Debierne  found 
still  another,  which  he  called  actinium. 

A  ton  of  pitchblende  contains  about  .17  gram  of  radium  and  ^oVu  as 
much  polonium.  Some  idea  of  the  labor  involved  in  extracting 
these  rare  elements  may  be  gained  from  the  fact  that  the  price  of 
radium  bromide  (the  form  in  which  radium  is  usually  obtained)  is 
quoted  at  $75,000  per  gram. 

The  value  of  radium  is  not  alone  due  to  its  use  in  scientific  research. 
Its  rays  have  been  found  to  be  a  cure  for  certain  disfigurements  and 
affections  of  the  skin,  accomplishing  in  many  cases  what  medicine 
and  surgery  can  not. 

508.  The  Three  Types  of  Rays.  —  The  rays  emitted  by 
the  radioactive  elements  are  invisible,  and  can  be  made  evi- 
dent and  investigated  only  by  their  effects.  Experiment 
has  shown  that  the  rays  are  of  three  kinds,  and  they  are 
designated  by  the  first  three  letters  of  the  Greek  alphabet, 
a  (alpha),  ft  (beta),  and  y  (gamma). 

The  alpha  rays  are  slightly  deviated  by  a  very  intense  magnetic 
or  electrostatic  field,  and  the  direction  of  the  deflection  shows  that 
they  carry  a  positive  charge;  they  are  very  efficient  ionizers,  as  shown 
by  the  rapid  discharge  of  an  electroscope;  they  excite  fluorescence, 
(see  spinthariscope,  below),  but  their  photographic  action  is  slight; 
they  are  completely  stopped  by  a  sheet  of  ordinary  writing  paper  or 
a  sheet  of  aluminum  .05  mm.  thick;  they  are  emitted  by  all  the  radio- 
active elements  named  above.  The  a  rays  consist  of  positively 
charged  particles  (ions),  shot  out  from  the  active  substance  within 


ELECTRICAL  THEORY  OF  MATTER     633 

average  jveiocity-jof  about  TJ^OQQ  mi.  per.  second—  The  charge  of  an 
a  particle  is  apparently  equal  to  that  of  the  hydrogen  ion  and  its 
mass  twice  as  great,  or  about  3400  times  the  mass  of 
an  electron. 

The  fluorescent  action  of  the  a  particles  is  beau- 
tifully shown  by  the  Spinthariscope  (Fig.  485),  a 
device  due  to  Sir  William  Crookes.  "It  is  formed  of 
a  short  brass  tube,  with  a  screen  coated  with  crystal- 
line zinc  sulphide  at  one  end  and  an  observing  lens  at 
the  other.  A  small  pointed  brass  needle  is  fixed  a  Fig.  485.— Spin- 
few  millimeters  above  the  screen,  and  on  the  side  of  thariscope.  S, 

f     fl  u  or  esce  n  t 
this  nearest  the  screen  a  very  minute  quantity  of     screen.    ^   ra_ 

radium  is  deposited  by  moistening  it  with  a  solution  dium  on  point 
of  a  radium  salt.  On  evaporation  an  invisible  film  of  needle- 
of  the  salt  remains."  Viewed  through  the  magnifying  lens,  the  screen 
presents  a  perfectly  dark  background,  upon  which  is  seen  a  multitude 
of  brilliant  points  of  greenish-white  light.  These  flash  out  and  disap- 
pear in  rapid  succession,  like  the  light  of  a  thousand  fireflies  on  a  dark 
night.  Each  flash  is  due  to  the  impact  of  an  a  particle,  and  is  given 
out  by  the  particular  crystalline  fragment  which  happens  to  be  struck. 
"  In  these  scintillations  we  have  possibly  the  only  direct  evidence  of 
the  action  of  one  individual  atom  known  to  science.  The  marvel  is 
most  impressive  when  it  is  remembered  that  the  effect  is  produced  and 
maintained  incessantly  by  a  quantity  of  radium  too  small  to  be  vis- 
ible, and  that  there  does  not  appear  to  be  the  slightest  loss  of  activity 
with  the  lapse  of  time."  —  C.  W.  Ra/ety. 

The  beta  rays  are  negatively  charged  particles,  for  they  are  strongly 
deflected  in  a  magnetic  or  electric  field,  and  the  deflection  is  opposite 
to  that  of  the  a  rays.  They  appear  to  be  identical  in  character  with 
the  cathode  rays,  i.e.  the  ft  particles  are  electrons.  Their  velocities 
vary  from  40,000  to  170,0x30  mi.  per  second.  Their  photographic 
action  is  strong,  their  ionizing  action  relatively  weak;  they  excite 
fluorescence.  Owing  to  their  high  velocity  they  have  considerable 
penetrating  power;  they  pass  readily  through  several  millimeters 
of  aluminum,  and  a  thickness  of  i  cm.  of  lead  is  required  to  stop  all 
of  them.  They  are  emitted  by  uranium,  radium,  thorium,  and  actin- 
ium, but  not  by  polonium. 

The  gamma  rays  are  always  found  in  association  with  the  ft  rays. 
They  have  all  the  properties  of  X-rays,  and  appear  to  be  of  the  same 


634  RADIATIONS 

nature,  i.e.  to  consist  of  pulses  in  the  ether,  traveling  with  the  velocity 
of  light.  It  is  believed  that  they  are  produced  by  the  expulsion  of 
the  ft  particles  from  the  active  substance.  They  have  great  pene- 
trating power.  Professor  Rutherford  states  that  the  y  radiation 
from  30  mg.  of  radium  could  be  detected  by  the  electroscope 
through  30  cm.  of  solid  iron. 


509.  Nature  of  Radioactivity.  —  According  to  all  chem- 
ical tests,  uranium,  thorium,  and  radium  are  elements; 
they  can  not  by  any  known  agency  be  decomposed  into 
unlike  constituents.  In  forming  compounds  with  other 
elements  they  exhibit  no  unusual  properties.  But  their 
radioactivity  is  something  wholly  different  from  chemical 
activities  in  general.  It  takes  place  spontaneously,  per- 
petually, and  can  neither  be  hastened  nor  retarded  by  any 
agency  known  to  man.  It  is  not  affected  in  the  slightest 
degree  by  the  cold  of  liquid  air  or  the  most  intense  heat. 
It  is  an  inherent  and  unalterable  property  of  the  atom  itself, 
persisting  unchanged  in  all  chemical  combinations  of  the 
radioactive  elements  with  other  forms  of  matter. 

The  student  who  is  familiar  with  elementary  chemistry 
will  readily  understand  that  radioactivity  involves  changes 
within  the  atom  itself,  changes  which  are  absolutely  foreign 
to  those  which  take  place  in  chemical  reactions.  With 
the  exception  of  the  gain  or  loss  of  a  few  electrons  (electric 
charges),  by  which  atoms  become  ions  and  vice  versa,  the 
atom  is  supposed  to  retain  its  identity  in  all  chemical 
changes  whatever.  The  loss  of  a  particle  of  atomic  size 
(the  a  particle)  from  an  atom  of  uranium,  thorium,  or 
radium  means  that  the  element  itself  is  changing  into  some- 
thing else,  —  a  thing  which  the  ancient  alchemists  sought 
to  accomplish  (the  transmutation  of  the  elements),  but 
which  the  modern  science  of  chemistry  found  to  be  impos- 
sible. It  is  impossible  by  human  agency,  so  far  as  is  known ; 


ELECTRICAL  THEORY  OF  MATTER     635 

but  the  facts  of  radioactivity  find  no  other  interpretation 
than  that  atomic  disintegration  is  constantly  going  on  in 
the  radioactive  elements,  as  a  spontaneous  process. 

610.  Disintegration   Products.     Origin   of  Radium.  —  An   enor- 
mous amount  of  experimental  work  has  been  done  to  determine 
the  nature  of  the  transformation  products  of  the  radioactive  elements, 
and  whole  volumes  have  been  written  on  the  subject.     In  such  work 
the  main  dependence  must  be  placed  on  the  evidence  afforded  by 
the  electroscope;  for  in  no  instance  have  these  products  been  obtained 
in  a  weighable  amount  except  from  the  ores,  in  which  they  have  been 
accumulating  during  geological  ages. 

Radium  continually  evolves  and  gives  off  a  gas  in  exceedingly 
minute  quantities.  This  gas  is  known  as  the  radium  emanation. 
It  is  intensely  radioactive  and  short-lived.  It  gives  rise  to  a  series 
of  disintegration  products  (eight  at  least),  which  are  solids  and  most 
of  which  are  radioctive.  What  is  supposed  to  happen  is  this:  The 
radium  atom  loses  an  a  particle  and  becomes  an  atom  of  the  emana- 
tion; the  latter  loses  an  a  particle  and  becomes  an  atom  of  the  solid 
deposit  from  the  emanation;  and  so  on  to  the  end.  In  some  of  the 
changes  /Sand  y  rays  are  also  given  out.  The  last  radioactive  product 
of  the  series  is  believed  to  be  polonium,  which  is  always  found  asso- 
ciated with  radium  in  pitchblende.  It  is  conjectured  that  polonium 
changes  into  lead.  The  a  particles  have  not  been  identified  with 
any  known  element.  Their  mass  seems  to  be  intermediate  between 
the  mass  of  the  hydrogen  atom  and  that  of  helium  (one  of  the  rare 
inert  gases).  They  may  possibly  be  atoms  of  the  latter  element;  for 
helium  is  found  occluded  in  the  radioactive  minerals. 

Radium  itself  appears  to  be  a  disintegration  product  of  uranium; 
it  is  always  found  in  fixed  proportion  to  the  amount  of  uranium  in 
uranium  minerals.  It  is  estimated  that  half  of  any  mass  of  radium 
will  disintegrate  in  the  course  of  2000  years.  It  is  confidently  ex- 
pected that  the  atomic  shower  in  a  spinthariscope  will  continue  dur- 
ing that  length  of  time,  with  a  loss  of  only  one  half  of  its  intensity. 

Thorium  and  actinium  give  rise  to  disintegration  products,  which 
form  a  definite  series  in  each  case,  as  with  radium. 

611.  Theory  of  Atomic  Disintegration.  —  The  disintegration  of 
an  atom  is  attended  by  the  liberation  of  energy.    The  amount  of 


636  RADIATIONS 

this  energy  is  no  less  astounding  than  the  facts  already  presented. 
Owing  to  the  high  velocities  of  the  a  and  /8  particles,  their  kinetic 
energy  is  relatively  very  great,  and  in  the  aggregate  is  seemingly 
out  of  all  proportion  to  the  mass  of  the  radioactive  substance.  This 
energy  is  directly  manifested  as  heat  in  any  compact  mass  of  radium; 
for  the  a  particles  shot  off  by  atoms  within  the  mass  are  stopped  by 
collisions  before  they  can  escape,  and  their  kinetic  energy  is  thus 
converted  into  heat. 

Now  it  is  found  that  radium  maintains  itself  at  a  higher  tempera- 
ture than  its  surroundings,  and  is  constantly  giving  out  heat  at  the 
rate  of  100  calories  per  gram  of  pure  radium  per  hour.  Computa- 
tion shows  that  the  total  heat  given  out  by  a  gram  of  radium  during 
its  "life"  amounts  to  something  like  3,000,000,000  calories,  which 
is  enough  to  raise  the  temperature  of  33  tons  of  water  from  the  freez- 
ing to  the  boiling  point.  If  this  energy  could  all  be  liberated  in  a 
single  explosion,  it  would  be  capable  of  blowing  a  2o,ooo-ton  battle- 
ship 230  ft.  into  the  air. 

Inconceivable  as  it  may  be,  we  are  forced  to  the  conclusion  that 
this  energy  is  stored  within  the  radium  atoms.  The  atom  is  pic- 
tured as  a  system  of  particles  in  rapid  orbital  motion.  Their  equilib- 
rium is  not  static,  but  dynamic,  like  that  of  the  solar  system.  When, 
from  whatever  cause,  the  atomic  system  becomes  unstable,  a  particle 
breaks  away  from  its  fellows  and  pursues  an  independent  course, 
with  the  velocity  it  chanced  to  have  at  the  instant. 

612.  The  Electron  Theory  of  the  Atom.  —  Theories 
concerning  the  constitution  of  .the  atom  are  still  in  the 
formative  state,  but  some  of  the  main  points  appear  to  be 
established.  Foremost  among  these  is  the  hypothesis 
that  the  electron  is  a  ronsti'tiient  of  all  atoms;  or,  rather, 
that-the-atomJs..  largely-made  up  of  electrons,  their  mim- 
bejrjmjijyTaiigemen t  varying jsvith  Jthe_diff erent. _dejnenls . 
These  electrons  are  in  rapid  oscillatory  or  orbital  motion, 
which  in  stable  atoms  (as  of  most  elements)  persists  through 
geological  ages,  perhaps  for  all  time,  like  the  revolution 
of  the  planets  round  the  sun. 

Heat  imparts  motion  to  the  molecule  as  a  whole,  and  pre- 


ELECTRICAL  THEORY  OF  MATTER     637 

sumably  also  to  its  constituent  atoms  and  to  the  electron 
systems  within  the  atoms.  An  oscillating  group  of  elec- 
trons (negative  electric  charges)  seems  to  answer  all  the 
requirements  of  a  source  of  ether  waves,  as  proposed  in 
the  electromagnetic  theory  of  light  (Art.  498).  The  charac- 
teristic spectra  of  the  different  elements  (Art.  493)  are 
accounted  for  on  the  supposition  that  different  kinds  of 
atoms  have  different  numbers  of  electrons,  differently 
arranged;  for  the  natural  oscillation  period  or  periods  of  a 
system  of  electrons  would  certainly  vary  with  their  num- 
ber and  grouping. 

One  of  the  greatest  difficulties  encountered  in  formulating  a  con- 
ception of  the  atom  is  in  reference  to  the  positive  electricity.  It  is 
doubtful  whether  this  has  ever  been  isolated  from  the  atom  as  the  elec- 
tron has.  The  positive  charge  of  the  atom  has  been  compared  by 
some  to  the  sun,  and  the  electrons  to  the  planets  revolving  about  it. 
Professor  J.  J.  Thomson,  who  is  the  leading  authority  in  such  matters, 
pictures  "a  sphere  of  uniform  positive  electrification"  with  the 
electrons  revolving  inside  it.  //  is  rather  startling  to  note  that, 
according  to  either  of  these  views,  matter  is  nothing  but  electricity. 

According  to  the  electron  theory,  positive  electricity,  being  insep- 
arable from  the  atom,  is  not  free  to  move  in  solid  conductors;  and 
an  electric  current  in  such  conductors  consists  simply  of  a  stream  of 
electrons,  moving  in  the  direction  opposite  to  what  is  universally 
termed  "the  direction  of  the  current."  Whatever  may  prove  to 
be  the  truth,  it  is  hardly  probable  that  the  present  conventional 
terminology  relative  to  the  electric  current  will  ever  be  changed. 


APPENDIX 

Table  I.    Metric  Units 

Deci-  means  tenth 

i  decimeter  (dm.)    =  .1  meter  (m.) 
i  decigram  (dg.)      =  .1  gram  (g.) 
Centi-  means  hundredth 

i  centimeter  (cm.)  =  .01  meter 
i  centigram  (eg.)     =  .01  gram 
Milli-  means  thousandth 

i  millimeter  (mm.)  =  .001  meter  =  .1  cm. 
i  milligram  (mg.)    =  .001  gram 
Kilo-  means  thousand 

i  kilometer  (km.)    =  1000  meters 
i  kilogram  (kg.)      =  1000  grams 

The  area  of  a  square  is  the  second  power  (square}  of  the  length  of 
one  side,     (i  sq.  ft.  =  i22  or  144  sq.  in.) 

i  square  centimeter  (sq.  cm.  or  cm.2)    =  100  sq.  mm. 
i  square  decimeter  (sq.  dm.  or  dm.2)     =  100  sq.  cm. 
i  square  meter  (sq.  m.  or  m.2)  =  100  sq.  dm. 

=  10,000  sq.  cm. 

The  volume  of  a  cube  is  the  third  power  (cube)  of  the  length  of 
one  side,     (i  cu.  ft.  =  i23  or  1728  cu.  in.) 

i  cubic  centimeter  (cu.  cm.  or  cm.3)      =  1000  cu.  mm. 
i  cubic  decimeter  (cu.  dm.  or  dm.3)       =  1000  cu.  cm. 
i  cubic  meter  (cu.  m.  or  m.3)  =  1000  cu.  dm. 

=  1,000,000  cu.  cm. 

Table  II.     Equivalents 
Metric  to  English  English  to  Metric 


i  cm. 

=  .3937  in. 

i  in. 

=  2.540  cm. 

i  m. 

=  39.37  in. 

i  ft. 

=  30.48  cm. 

=  3.281  ft. 

i  yd. 

=  .9144  m. 

i  km. 

=  .6214  mile 

i  mile 

=  1.6093  km. 

638 

APPENDIX 


639 


i  cm.2 

=  .i55osq.  in. 

i  sq.  in. 

=  6.452  cm.2 

im.2 

=  1.196  sq.  yd. 

i  sq.  ft. 

=  929.0  cm.2 

=  10.764  sq.  ft. 

i  sq.  yd. 

=  .8361  m.2 

i  cm.3 

=  .06103  cu.  in. 

i  cu.  in. 

=  16.387  cm.3 

i  dm.3 

=  1.0567  qt.  (liquid) 

I  CU.  ft. 

=  28,315  cm.3 

i  cu.  m.3 

=  1.308  cu.  yd. 

i  cu.  yd. 

=  .7645  m.3 

=  35-3I7  cu.  ft. 

i  qt.' 

=  .9463dm.3  (liters) 

i  gal. 

=  3.785  liters 

i  gram 

=  .0353  oz. 

I  OZ. 

=  28.35  g- 

i  kg. 

=  2.2046  Ib. 

ilb. 

=  453-6  g. 

Table  III.     Mensuration  Rules 

ratio  of  the  circumference  of  a  circle  to  its  diameter  =  3.1416 

Circumference  of  a  circle  (radius  r)  =•  2  irr 
Area  of  a  circle  =  trr2 

Surface  of  a  sphere  =  4  irr2 

Volume  of  a  sphere  =  t  irr3 

Lateral  surface  of  a  right  cylinder 

(altitude  h  and  radius  of  base  r)    =  2  irrh 
Volume  of  a  right  cylinder  =  irr2/* 

Table  IV.     Densities  (in  grams  per  ccm.) 
Solids  (Except  Mercury) 


Aluminum,  cast  . 

2.58 

Iron,  bar  . 

7-8 

Antimony,  cast    . 

6.72 

Iron,  cast 

7.2  to    7.3 

Beeswax    . 

.96 

Ivory 

1.9 

Bismuth,  cast 

9.8 

Lead    .     .  •  -„    ...  j 

11.3  to  11.4 

Brass  .... 

8-5 

Marble     .      .      . 

2.72 

Copper     . 

8.8  to  8.9 

Mercury,  at  o°  C. 

13-596 

Cork    .      .      .     : 

.14  to  .24 

Platinum  . 

21.5 

Galena 

7.58 

Quartz 

2:65 

German  silver 

8-5 

Silver  .      .      .  .  . 

10.4  "to  10.5 

Glass,  crown  . 

2-5 

Steel    .... 

7.8  to   7.9 

Glass,  flint     .      . 

3     to     3.5 

Sulphur,  native   . 

2.03 

Gold    .... 

iQ-3 

Tin      .... 

7-3 

Ice. 

.017 

Zinc,  cast 

7.1 

640 


APPENDIX 


Liquids  and  Solutions 


Alcohol  (95%)      . 
Blood  .... 
Carbon  disulphide 
Chloroform     . 
Copper  sulphate  so- 
lution    . 

Ether  .... 
Glycerine  . 
Hydrochloric  acid 
Mercury  at  o°  C. 


.82 

i.o6 
1.29 


1.16 
.736 
1.27 

1.22 
I3-596 


Milk       .        .        . 

Nitric  acid 

Oil  of  turpentine 

Olive  oil 


1.03 

i-5 
.87 
-915 


Salt  solution  (NaCl), 

saturated     .     .       1.205 
Sulphuric  acid  (15%)  i.io 
Sulphuric  acid      .       1.8 
Water  (4°  C.)       .       i.ooo 
Water,  sea  1.026 


Gases  at  o°  C.  and  76  cm.  Pressure 


Air 001293 

Carbon  dioxide    .         .001977 
Carbon  monoxide        .001250 


Hydrogen 

Nitrogen 

Oxygen 


.0000896 

.001257 

.001429 


INDEX 


THE  NUMBERS  REFER  TO  PAGES 


Aberration,  chromatic,  443 

spherical,  387,  414 
Absolute  temperature,  245 
Absolute  units  of  force,  126 
Absorption,  of  radiation,   231,   235, 

237 

selective,  237 
Acceleration,  109-119 

due  to  gravity,  no,  115-119 
Achromatic  lens,  444 
Action  and  reaction,  14,  129-131,  144 
Adhesion,  200 
Aeroplanes,  191-193 
Air,  buoyancy  of,  59 

composition  of,  7,  266 

density  of,  43 

water  vapor  in,  266-272 
Air  pump,  61 
Amalgamating  zinc,  512 
Ammeter,  536 
Ampere,  532,  605 
Angle,  critical,  401 

of  deviation,  390 

of  incidence  and  reflection,  374 

of  refraction,  390 

refracting,  of  prisms,  398 

sine  of,  394 

visual,  419-420 
Anode,  602 
Antinode,  348 
Arch,  98 

Archimedes,  principle  of,  37 
Armature,  of  dynamos  and  motors, 

p  573 

Artesian  wells,  31 
Atmosphere,  heating  of,  237 

height  of,  57-58 
Atmospheric  electricity,  497-499 


Atmospheric  pressure,  43-51 

refraction,  402-405 
Atom,  198 

nature  of,  613,  636 
Atomic  disintegration,  635 
Attraction,  electrostatic,  478 

magnetic,  457-458 

molecular,  199-201 

of  gravitation,  141-143 
Audibility,  limits  of,  328 
Aurora  borealis,  58,  624 
Axis,  of  lens,  407,  409 

of  mirror,  379 

• 

Balance,  19 
Balloon,  66,  191-193 
Barometer,  47-51 
Battery,  see  Cells 
Beam,  of  light,  364 
Beats,  331-333 
Bell,  electric,  524 
Bellows,  63 
Bichromate  cell,  514 
Biograph,  435 
Boiling,  273-275 
Boiling  point,  223,  274 
Boyle's  law,  55 
Buoyancy,  center  of,  89 

of  air,  59 

of  liquids,  36-38 

Caloric  theory,  218 
Calorie,  248 
Calorimetry,  248-251 
Canal  rays,  627 
Capillarity,  211-214 
Capstan,  169 
Cathode,  602 


641 


642 


INDEX 


Cathode  rays,  624-627 
Cells,  electric,  503-517 

electro-motive  force  of,  538 

in  battery,  549~552 

storage,  607 
Center  of  buoyancy  89 

of  curvature,  379 

of  gravity,  84-86 
Centrifugal  force,  139 
Centripetal  force,  135-139 
Charge,  electrostatic,  579,  589 
Charles,  law  of,  244 
Chemical  changes,  4 

effects  of  electric  current,  600-608 

energy,  180,  289 
Chromatic  aberration,  443 

scale,  328 
Circuit,  electric,  510 

divided  or  shunt,  543 
Clouds,  271 

Coal,  energy  of,  289-290 
Cohesion,  199-201 
Cold  storage,  280      , 
Color,  441,  447-455 

by  interference,  454 

vision,  theory  of,  453 
Commutator,  578 
Compass,  476 
Compressibility,  of  gases,  8,  204 

of  liquids  and  solids,  8,  208,  215 
Compression  pump,  62 
Condensation,  of  gases,  279 

of  water  vapor,  271 
Condenser,  electric,  495 
Conduction  of  heat,  225-227 
Conductivity,  electric,  542 
Conservation,  of  energy,  181,  288,  290 
Convection,  of  heat,  227-229 
Couple,  82,  90 
Critical  angle,  401 

temperature,  279 
Crook es  tubes,  624 
Current,  electric,  chemical  effects  of, 
601-608 

danger  from,  598 

heating  effects  of,  556-563 

induced,  564-571 

magnetic  effects  of,  517-531 

measurement  of,  53J-537 


Current,  nature  of,  637 

sources  of,  502 

unit  of,  532,  605 
Curvilinear  motion,  135-140 

Dalton's  laws,  264 

Daniell  cell,  517 

D' Arson val  galvanometer,  535 

Declination,  magnetic,  472 

Density,  20 

and  pressure  of  gases,  57 
Derrick,  77,  170 
Deviation,  angle  of,  390 
Dew,  271 
Dew-point,  267 
Diffusion,  of  gases,  202 

of  light,  373 

of  liquids,  207 
Dipping  needle,  474 
Discord,  326,  332,  341 
Dispersion,  of  light,  437-442 
Distillation,  275 
Diving  bell,  63 
Divisibility,  of  matter,  197 
Dynamics,  104-193 

definition  of,  104 

of  fluids,  183-193 

of  solids,  104-150 
Dynamos,  573-588 
Dyne,  126 

Ear,  356 

Earth,  effect  of  rotation  on  its  shape, 
144-145 

effect  of  rotation  on  weight,  145 

magnetic  field  of,  472-476 

revolution    and    rotation    of,    19, 

141-142 
Echoes,  320 
Eclipses,  365 
Edison,  371,  595,  608 
Efficiency,  of  machines,  161 
Elasticity,  93 
Electric,  arc,  559 

battery,  549~552 

bell,  524 

cells,  503-51 7 

circuit,  510 

conductors  and  non-conductors,  479 


Electric  cooking  and  heating,  561 

current,  see  Current 

discharge,  in  rarefied  gases,  622- 
627- 

energy,  533,  553-556,  588 

forging  and  smelting,  561 

fuses,  562 

light,  558-561 

measurements,  531-553 

motors,  583-587 

oscillations,  617 

potential,  see  Potential 

power,  533,  555 

resistance,  see  Resistance 

spark,  494 

telegraph,  525-531 

transmission  of  power,  589-592 
Electricity,  current,  501-608 

atmospheric,  497-499 

nature  of,  Chap.  XIV 
Electrification,  478-485 
Electrodes,  505,  602 
Electrolysis,  600-605 
Electrolytic  dissociation,  508,  603 
Electro-magnet,  520-523 
Electro-magnetic  field,  517-520 

induction,  564-571 

theory  of  light,  619 

waves,  618 

Electro-metallurgy,  606 
Electro-motive  force,  537 

measurement  of,  547 

of  cells,  538 

unit  of,  537 

Electrons,  625,  633,  636 
Electrophorus,  487 
Electroplating,  605 
Electroscope,  481 
Electrostatics,  477-500 
Electrostatic  attraction   and   repul- 
sion, 478 

capacity,  497 

charge,  479,  489 

condenser,  495 

field,  499 

induction,  482 

machines,  486-488 

potential,  490-494 
Electrotyping,  605 


INDEX  643 

Energy,  availability  of,  182 
Energy,  conservation  of ,  181,  288,  290 
dissipation  of,  182 
forms  of,  chemical,  180,  289 
electrical,  533,  553~556 
kinetic,  151-157 
mechanical,  177 
molecular    kinetic    (heat),    206, 

218-220 
molecular     potential      ("  latent 

heat"),  258-259,  276-278 
muscular,.  180 
of  light,  361,  443 
of  rotation,  178 
of  sound,  315-317 
potential,  177,  180 
radiant,  230-239,  290 
solar,  289-291 
sources  of,  160,  289 
transformation  of,  by  combustion, 

289 
by  compression  and  expansion 

of  gases,  278 
by  dynamos,  582 
by  electrical  resistance,  556 
by  friction,  179 

by  fusion  and  solidification,  258 
by  motors,  585 
by    radiation    and    absorption, 

230,  235-237 

by  steam  engine,  293,  297 
by  vaporization  and  condensa- 
tion. 264,  276 
transmission  of,  by  electric  current, 

511,  588-592 
by  machines,  159-162,  166,  i6P 

172 

units  of,  electrical,  556 
mechanical,  154 
thermal,  248 
Engine,  steam,  292-299 
compound,  295 
condensing,  294 
efficiency  of,  297 
steam  turbine,  302-304 
gas,  299-302 
gasoline,  302 

Equilibrium,    of    concurrent    forces, 
69-76 


644 


INDEX 


Equilibrium,  of  floating  bodies,  89 

of  parallel  forces,  78-79 

of  two  forces,  69 

stable,     unstable,     and     neutral, 

84-90 
Ether,  luminiferous,   233,  360,  471, 

618 
Evaporation,  260-267 

cooling  by,  264,  281 
Expansion,  by  heat,  206,  208,  215, 
240-245 

cooling  by,  278 
Extension,  16 
Eye,  416-425 

care  of,  422 

defects  of,  420 

Falling  bodies,  no,  115-118 

Faraday,  564,  605 

Far  sight,  421 

Field,  electro-magnetic,  517-520 

electrostatic,  499 

magnetic,  466-471 
Field  magnet,  573 
Floating  bodies,  buoyancy  upon,  38 

equilibrium  of,  89 
Fluids,  characteristics  of,  7-9 

dynamics  of,  183-193 
Flywheel,  178 
Focal  length,  of  lens,  407 

of  mirrors,  382 
Foci,  of  lenses,  407-410 

of  mirrors,  379-384 
Fog,  271 
Foot-pound,  154 
Force,  9-12 

buoyant,  36-38,  59 

centrifugal,  139 

centripetal,  135-139 

electro- motive,  537 

elements  of,  70 

graphic  representation  of,  70 

moments  of,  80-83 

resultant,  13 

units  of,  1 8,  126 
Forces,  balanced,  12,  68-79 

composition  of,  73 

concurrent,  69-76 

molecular,  199-201,  204 


Forces,  parallel,  78-79 

parallelogram  of,  74 

resolution  of,  75 

unbalanced,  12,  120-149 
Force  pump,  64 
Franklin,  498 
Fraunhofer  lines,  440 
Freezing,  253-255 
Freezing  mixtures,  259-260 
Freezing  point,  223,  256 
Friction,  10 

heating  effects  of,  179,  218-220 

uses  of,  131,  174-175 
Frost,  271 
Fulcrum,  163 

Fundamental  tone,  336-342,  351-354 
Fusion,  252-259 

change  of  volume  during,  255 

heat  of,  257-259 

Galilean  telescope,  432 
Galileo,  48,  131,  433 
Galvanometers,  533-537 
Gas  engines,  299-302 
Gases,  characteristics  of,  7-9 

compressibility  of,  8,  204 

cooled  by  expansion,  278 

diffusion  of,  202 

distinguished  from  vapors,  9,  262- 
264,  279 

effect  of  pressure  on  volume  and 
density  of,  55-57 

effect  of  temperature  on  volume  of, 
206,  244 

kinetic  theory  of,  207 

liquefaction  of,  279 

molecular  properties  of,  202-207 

pressure  of,  52-57 

statics  of,  43-67 
Gasoline  engines,  302 
Gay-Lussac,  law  of,  244,  246 
Geissler  tubes,  623 
Glaciers,  flow  of,  257 
Gram,  mass  and  weight,  18 
Gram-centimeter,  154 
Gravitation,  141-145 

law  of,  143 
Gravity,  84,  145 

acceleration  due  to,  no,  115-118 


INDEX 


645 


Gravity  cell,  516 

center  of,  84-88 

pressure  due  to,  24-32 

specific,  39-41 
Guericke,  44,  6 1 

Hail,  272 

Harmony,  326,  341 
Hearing,  356-358 
Heat,  179,  206,  218-304 

conduction  of,  225-227 

convection  of,  227-229 

expansion  due  to,   206,  208,  215, 
240-245 

kinetic  theory  of,  206,  218-220 

mechanical  equivalent  of,  287 

of  fusion,  257 

of  vaporization,  276, 

sources  of,  289 

specific,  248-251 

unit  of,  248 
Heat  engines,  292-304 
Heating  of  buildings,  283-287 
Helmholtz,  291,  356 
Hooke's  law,  94 
Horse-power,  157 
Humidity,  268 
Hydraulic  press,  34 
Hygrometer,  270 

Ice,  253-258 

manufacture  of,  280 
Illumination,  intensity  of,  368 

artificial,  402 
Images,  by  lens,  409-414 

by  plane  mirrors,  375-378 

by  small  opening,  367 

by  spherical  mirrors,  382-388 

real,  380,  383,  409,  412 

virtual,  385,  410,  413 
Incandescent  lamp,  557-558 
Inclined  plane,  76,  112,  173 
Index  of  refraction,  395 
Induced  currents,  564-571 
Induction,  earth's,  474 

electro-magnetic,  564-571 

electrostatic,  482 

magnetic,  460 

self,  569 


Induction  coil,  570-572 
Inertia,  9-11,  120,  125 
Insulators,  electric,  479,  590 
Interference,  of  light,  454 

of  sound,  329-332 
Ions,  508,  603-604,  627,  630 
Iridescence,  455 

Joule,  220,  288 
Joule's  equivalent,  287 
law,  556 

Kilogram-meter,  154 
Kinetic  energy,  151-157 
Kinetic  theory,  of  gases,  202-207 

of  heat,  206,  218-220 
Kinetics,  see  Dynamics 

Lamp,  arc,  559 

incandescent,  557-558 
Lantern,  optical,  435 
Law,  Boyle's,  55-57 

Dalton's  264 

Hooke's,  94 

Joule's,  556 

of  Charles,  244 

Ohm's,  539 

Pascal's,  33 

Laws  of  motion,  Newton's,  120-132 
Laws  of  nature,  28 
Leclanche  cell,  514 
Lenses,  achromatic,  443-445 

concave,  413 

convex,  405-413 
Lever,  163 
Leyden  jar,  495 
Lifting  pump,  63 
Light,  359-456 

dispersion  of,  437-442 

intensity  of,  369-371 

interference  of,  454 

propagation  of,  362-363,  366 

reflection  of,  372-375^ 

refraction  of,  389-398 

theory  of,  359-361,  619 

velocity  of,  362 

wave  length  of,  366,  441 
Lightning,  497 

rod,  499 


646 


INDEX 


Lines  of  force,  466-471,  576 

Lintel,  97 

Liquids,  characteristics  of,  7-9 

diffusion  of,  207 

dynamics  of,  183-193 

molecular  properties  of,  207-214 

statics  of,  22-42 
Liter,  16 

Local  action  in  voltaic  cell,  511 
Locomotive  engine,  296 
Lodestone,  457 
Loudness  of  sound,  314 

Machines,  159-175 

efficiency  of,  161 

electrical,  486-488 

mechanical  advantage  of,  161 
Magdeburg  hemispheres,  44 
Magnetic  declination,  472 

effects  of  a  current,  517-531 

field,  466-471 

inclination  or  dip,  473 

induction,  460 

lines  of  force,  466 

meridian,  472 

needle,  458 

permeability,  462 

poles,  458 

substances,  460 
Magnetism,  457-476 

terrestrial,  472-476 
Magnetization,  permanent  and  tem- 
porary, 461    . 

theory  of,  462-465 
Magnifying  glass,  426 

power,  426 
Major  triad,  326 
Manometers,  53-55 
Mass,  center  of,  84 

definition  of,  17 

measurement  of,  by  weight,  19 

units  of,  1 8 
Matter,  divisibility  of,  197 

properties  of,  7-9,  194-217 

states  of,  7-9,  200,  252 

structure  of,  196-202 
Measurement,  15-20 
Mechanical  advantage,  161 

equivalent  of  heat,  287 


Mechanics,  definition  of,  5,  22 
Melting  points,  254 

effect  of  pressure  on,  256 
Meter,  16 
Microphone,  595 
Microscope,  compound,  427 

simple,  426 
Mirage,  404 
Mirrors,  parabolic,  387 

plane,  376-378 

spherical,  379~3S& 
Molecular  forces,  199-201,  204 

motion,  202-209,  215 

structure  of  matter,  194-217 
Molecule,  197 
Moment  of  force,  80-83 
Momentum,  127 
Moon,  revolution  of,  143 
Motion,  104-120 

accelerated,  108 

curvilinear,  135-140 

laws  ofr  120-132 

of  falling  bodies,  115-118 

of  pendulum,  146-149 

of  projectiles,  116-118 

on  an  inclined  plane,  111-114 

uniform,  105 

wave,  309 

Motor,  electric,  583-587 
Musical  instruments,  344,  350-354 

intervals,  324-328 

scales,  326-328 

sounds,  313,  324 

Newton,  132,  142 
Newton's  law  of  cooling,  236 

laws  of  motion,  120-132 
Nodes,  337,  347 
Noise,  324 

Octave,  325 

Ohm,  definition  of,  538    \*S 

Ohm's  law,  539    , 

Opera  glass,  432 

Optical  instruments,  425-436 

Organ  pipes,  35°-3S3 

Overtones,  336-341,  35i~354 

Parallelogram  of  forces,  74 


INDEX 


647 


Pascal's  law,  33 

Pendulum,  146-149 

Penumbra,  365 

Permeability,  magnetic,  462 

Phenomena,  natural,  5 

Phonograph,  344 

Photometry,  369-371 

Physical  changes,  4 

Pinhole  image,  367 

Pitch,  of  musical  sounds,  307,  322, 

324-329 
Plasticity,  94 

Polarization,  in  voltaic  cell,  512 
Poles,  magnetic,  458 
Porosity,  214 
Potential,  electric,  490 

fall  of,  540 

Potential  energy,  177,  180 
Pound,  weight  and  mass,  18 
Poundal,  127 
Power,  157 

electric  transmission  of,  589-592 
Pressure,  atmospheric,  43-51 

of  gases,  52-57 

of  liquids,  24-36 

of  moving  fluids,  183-187 

of  vapors,  262-264,  274 
Pressure  gauges,  53-55 
Prism,  398 
Prism  binocular,  433 
Projectiles,  116-118 
Properties  of  matter,  194-217 
Pulleys,  170-173 
Pump,  air,  61 

compression,  62 

force,  64 

steam,  65 

suction,  63 

Quality  of  sound,  322,  338-341 

Radiant  energy,  230-238,  361 

emission  of,  231,  235 

reflection  of,  236 

selective  absorption  of,  237 

transmission  of,  230-234 

visible  and  invisible,  232 
Radioactivity,  630-637 
Radiometer,  235 


Radium,  631,  634 

Rain,  272 

Rainbow,  445 

Ray,  of  light,  364 

Reaction  and  action,   14,   129-131, 

144 
Reflection,  of  light,  372-375 

of  sound,  320 

total,  399-401 
Refraction,  389-398 

atmospheric,  402-405 

index  of,  395 

laws  of,  393 

relation  to  velocity,  390-395 
Relay,  telegraph,  527 
Resistance,  161 

of  the  air,  123,  189 

of  water,  189 
Resistance,  electrical,  538 
laws  of,  541-545 
measurement  of,  546,  548 
specific,  542-543 
unit  of,  538 
Resistance  coils,  546 
Resolution,  of  a  force,  75 

of  a  velocity,  106 
Resonance,  346-350 
Resultant  force,  13,  74,  79 

velocity,  106 
Rumford,  Count,  218,  369 

Scales,  musical,  326-328 

Screw,  174 

Screw  propeller,  190 

Selective  absorption,  237 

Self-induction,  569 

Shadows,  364 

Short  sight,  420 

Shunt  circuit,  543 

Sine  of  an  angle,  394 

Siphon,  65 

Sky,  color  of,  450 

light  of,  373 
Snow,  272 
Soap  bubbles,  210 
Solenoid,  520 
Solidification,  252-258 
Solids,  characteristics  of,  7,  8 

dynamics  of,  108-183 


648 


INDEX 


Solids,  molecular  properties  of,  214- 
217 

statics  of,  68-103 
Solution,  heat  of,  259 
Sonometer,  333 
Sound,  305-358 

intensity  of,  314-3*7 

interference  of,  329-332 

loudness  of,  314 

media,  308 

origin  of,  306 

pitch  of,  307,  322,  324-329 

properties  of,  322-342 

quality  of,  322,  338-341 

reflection  of,  320 

transmission  of,  308-314 

velocity  of,  317-319 

waves,  311,  329 
Sounder,  telegraph,  526 
Speaking  tubes,  317 
Specific  gravity,  39-41 

heat,  248^251 

resistance,  542 
Spectrum,  438-440 

invisible,  442 

solar,  615 
Spectra  and  spectrum  analysis,  610- 

616 

Speed,  105 

Spherical  aberration,  387,  414 
Stability,  88-90 
Stars,  distance  of,  362 

twinkling  of,  403 
Statics  of  gases,  Chap.  IV 

of  liquids,  Chap.  Ill 

of  solids,  Chap.  V 
Steam,  heating  by,  286 

pressure  of,  275 

saturated,  262 

superheated,  262,  298 
Steam  engines,  292-299 
Stereoscope,  424 
Stresses  and  strains,  92-102 
Strings,  vibration  of,  334-341 
Sun,  energy  of,  289-291 
Surface  tension,  210-212 
Suspension  cable,  102 
Sympathetic    vibrations,    342,   345- 
350 


Tangent  galvanometer,  533 
Telegraph,  525-531 
Telegraphy,  wireless,  620 
Telephone,  592-598 

acoustic,  317 

exchange,  597 
Telescopes,  429-434 
Temperature,  220-224 

absolute,  245  ~~ 
Tenacity,  94,  217 
Terrestrial  magnetism,  472-476 
Theory,  definition  of,  195 

of  electricity,  630-637 

of  gases,  202-207 

of  heat,  206,  218-220 

of  light,  359-361,  619 

of  magnetic  action,  470 

of  magnetization,  462-465 

of  the  structure  of  matter,  194-217 
Thermometers,  222-224,  242 
Thrust,  30 
Thunder,  497 
Time,  unit  of,  19 
Tone,  324 

fundamental,  336-342 
Torricelli,  49 

Total  reflection,  399-401^ 
Transference  of  energy,  see  Energy 
Transformation  of  energy,  see  Energy 
Transformer,  590-592 
Truss,  loo-ioi 

Tuning  fork,  vibration  of,  307 
Turbine  engines,  302-304 

Umbra,  365 

Units,  fundamental,  20 

of  acceleration,  no 

of  current  strength,  532,  605 

of  electrical  power,  555 

of  electrical  resistance,  538 

ofE.M.F.,  537 

of  extension,  16 

of  fluid  pressure,  29,  55 

of  force,  1 8,  126-127 

of  heat,  248 

of  mass,  1 8 

of  mechanical  power,  157 

of  time,  19 

of  velocity,  105 


INDEX 


649 


Units  of  work  and  energy,  154 


Vacuum,  47 

Vapor,  atmospheric,  266-272 

pressure  of,  262-264 
Vaporization,  260-282  - 

heat  of,  276 

Vapors,  9,  262-264,  279 
Velocity,  105-108 

graphic  representation  of,  105 

resolution  of,  106 

uniform,  105 

variable,  108-119 
Velocities,  composition  of,  105 
Ventilation,  283-287 
Vibration,  forced    and    sympathetic, 
342-358 

of  air  columns,  346-354 

of  bells,  342 

of  pendulum,  147 

of  strings,  334~34i 
Vision,  binocular,  423-425 
Vocal  cords,  355 
Voice,  354 
Volt,  537 
Voltaic  cell,  507 
Voltmeter,  547 


Water,  compressibility  of,  8,  208 

density  of,  20 

electrolysis  of,  502 

evaporation  of,  265-267 

expansion  of,  243 
Water  wheels,  185-188 
Watt,  555  < 
Wave  motion,  309 
Waves,  of  light,  359~36i 

of  sound,  311,  329 

of  water,  310 

Weather,  prediction  of,  50 
Wedge,  174 
Weighing,  19 
Weight,  11-12,  16  „ 
Welding,  200 

electric,  561 
Wheel  and  axle,  167 
Wind  instruments,  350-354 
Windlass,  168,  170 
Winds,  229 
Work  and  energy,  150-157 

units  of,  154 

X-rays,  628 
Zero,  absolute,  245 


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